Perl 5 + -M5.10.0, 77 bytes

This solution only has a 256 colour palette.

sub x{".[48;5;@{[rand>.5?$-||=rand 254:254]}m "}say$}=x,$b=x,x,$b,$}for 0..4

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Explanation

Nothing particularly clever except perhaps:

• Use $- to automatically round the colour number for use in the escape sequence, used instead of $n=0|rand 254.

MATL, 36 29 bytes

5l$rtP+!kllII$r*O16tQ/XE'a'YG

This saves the result in file a.png.

Replacing 'a' by 2 in the code displays the image (scaled up) instead of saving a file:

5l$rtP+!kllII$r*O16tQ/XE2YG

Here's an example output:

Explanation

5l$r     % 5×5 matrix of independent random values with uniform distribution
         % on the interval (0,1)
tP+!     % Duplicate, flip vertically, add, transpose. This gives a horizontally
         % symetric matrix. Center column pixels are uniformly distributed on the 
         % interval (0,2). Rest have a triangular distribution on (0,2)
k        % Round down. In either of the above cases, this gives 0 and 1
         % with the same probability
llII$r   % 1×1×3 array of independent random numbers with uniform distribution
         % on (0,1). This is the foreground color.
*        % Multiply the two arrays with broadcast. Gives a 5×5×3 array. Ones in the
         % 5×5 array become the random foreground color. Zeros remain as zeros.
O        % Push 0
16tQ/    % 16, duplicate, add 1, divide: gives 16/17, or 240/255
XE       % Replace 0 by 16/17: background color
'a'      % Push file name
YG       % Write image to that file

Actually, 53 bytes

3'≡*;╗`┘#8╙r-J┌`MΣ╝5`3"2rJ└"£n3╟;RdX+Σ`nkΣ"P6 5 5 255

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Actually is bad at string processing. I think I've mentioned that before. This program outputs a P6 netpbm image using CP437, like the following:

P6 5 5 255
εεεεεεå♠ƒεεεεεεå♠ƒå♠ƒεεεå♠ƒå♠ƒεεεεεεεεεεεεεεεεεεεεεå♠ƒεεεεεεå♠ƒå♠ƒå♠ƒå♠ƒå♠ƒ

This can be converted to a PNG using ImageMagick:

seriously -f prog.srs | convert - out.png

Scaled up PNG version:

Explanation:

3'≡*;╗`┘#8╙r-J┌`MΣ╝5`3"2rJ└"£n3╟;RdX+Σ`nkΣ"P6 5 5 255
3'≡*                                                   push "≡≡≡" (char 240, 3 times)
    ;╗                                                 save a copy to reg0
      `┘#8╙r-J┌`M                                      map:
       ┘#                                                cp437 char code, as list ([240])
         8╙r-J                                           random value in range(2**8) that is not 240
              ┌                                          char code
                 Σ╝                                    concatenate, push to reg1
                   5`3"2rJ└"£n3╟;RdX+Σ`n               do this 5 times:
                     3"2rJ└"£n                           do this 3 times:
                       2rJ└                                randomly pick 0 or 1, push the value in that register
                              3╟                         push the top 3 items to a list
                                ;RdX+                    duplicate, reverse, discard first value, append (mirror the list)
                                     Σ                   concatenate
                                       kΣ              push stack as list, concatenate
                                         "P6 5 5 255   prepend header

Python, 167 164 155 148 bytes

Output as ppm

from random import*
print"P3 5 5 255"
C,P=choice,[[240]*3,[randint(0,255)for _ in"RGB"]]
exec"a,b=C(P),C(P);print' '.join(map(str,a+b+C(P)+b+a));"*5

• Edit1: range(5) to " "*5
• Edit2: shoved a byte at the first print
• Edit3: Replaced list comprehension with map since only str(p) was used
• Edit4: exec""*5 instead for _ in " "*5:

Some improvement over old, string-based code:

from random import*
print "P3 5 5 255"
j,c,a=' '.join,choice,['']*5
P=["240 "*3,j([str(randint(0,255))for _ in"RGB"])]
for _ in a:a=[c(P)]*5;a[1]=a[3]=c(P);a[2]=c(P);print j(a)

Mathematica, 105 102 98 94 bytes

c=1~RandomReal~3;r@l_:={c,4{4,4,4}/17}~RandomChoice~{l,5};Image[Join[#,r@1,Reverse@#]&@r@2]

 is the Transpose operator.

Edit 1: Saved 3 bytes by replacing Round + RandomReal with RandomInteger

Edit 2: Saved 4 bytes by replacing RandomInteger with RandomChoice

Edit 3: Saved 4 bytes by replacing RandomColor and GrayLevel with RandomReal

MATLAB, 102 bytes

Hi, this is my matlab solution:

p=@(x)(imwrite(floor(randi(2,5,2)*[eye(2) ones(2,1)./2 fliplr(eye(2))]),[0.9412*[1 1 1];rand(1,3)],x))

The input x of the function is the name of the output file. For example:

p('output.png')

produces a png image called 'output.png'.

Here the results of some executions of this code.

Swift 2.3, 593 585 bytes

var t = 0,g = UIGraphicsGetCurrentContext(),c = UIColor(hue:CGFloat(drand48()),saturation:1,brightness:1,alpha:1).CGColor
srand48(time(&t))
UIGraphicsBeginImageContext(CGSizeMake(5,5))
for x in 0..<3 {for y in 0..<5 {CGContextSetFillColorWithColor(g,drand48()>0.5 ? c : UIColor.whiteColor().CGColor)
var r = [CGRect(x:x,y:y,width:1,height:1)]
if x<2 {let m = x==0 ? 4 : 3;r.append(CGRect(x:m,y:y,width:1,height:1))}
CGContextFillRects(g,&r,r.count)}}
let i = UIGraphicsGetImageFromCurrentImageContext()
UIImagePNGRepresentation(i)!.writeToURL(NSURL(string:"/a/a.png")!,atomically:true)

Update

Swift 3, 551 bytes

var t = 0,g = UIGraphicsGetCurrentContext()!,c = UIColor(hue:CGFloat(drand48()),saturation:1,brightness:1,alpha:1).cgColor
srand48(time(&t))
UIGraphicsBeginImageContext(CGSize(width:5,height:5))
for x in 0..<3 {for y in 0..<5 {g.setFillColor(drand48()>0.5 ? c : UIColor.white().cgColor)
var r = [CGRect(x:x,y:y,width:1,height:1)]
if x<2 {let m = x==0 ? 4 : 3;r.append(CGRect(x:m,y:y,width:1,height:1))}
g.fill(&r,count: r.count)}}
let i = UIGraphicsGetImageFromCurrentImageContext()!
try!UIImagePNGRepresentation(i)!.write(to: URL(string:"/a/a.png")!)

I'm at WWDC and just got the Xcode 8 beta with Swift 3. Apple made some of the CoreGraphics calls more "Swifty," and I am able to reduce the bytecount.

Swift 2 code Ungolfed:

var t = 0
srand48(time(&t))

UIGraphicsBeginImageContext(CGSizeMake(5,5))
let context = UIGraphicsGetCurrentContext()
let color = UIColor(hue: CGFloat(drand48()),saturation:1,brightness:1,alpha:1).CGColor

for x in 0..<3 {
    for y in 0..<5 {
        CGContextSetFillColorWithColor(context, drand48() > 0.5 ? color : UIColor.whiteColor().CGColor)
        var rects = [CGRect(x:x,y:y,width:1,height:1)]

        if x < 2 {
            let mirror = x==0 ? 4 : 3
            rects.append(CGRect(x: mirror, y: y, width: 1, height: 1))
        }

        CGContextFillRects(context, &rects, rects.count)
    }
}


let image = UIGraphicsGetImageFromCurrentImageContext()
UIImagePNGRepresentation(image)!.writeToURL(NSFileManager.defaultManager().URLsForDirectory(.DocumentDirectory,inDomains:.UserDomainMask).first!.URLByAppendingPathComponent("a.png"), atomically:true)

This answer assumes UIKit is available and uses the Cocoa Touch framework.

Some example output images:

I know I can't compete with most of the other answers but I wanted to give this a shot as a personal challenge. There is definitely room for improvement with this answer, but I think it will be hard to get this down below a few hundred bytes due to the length of the UIKit and CoreGraphics image writing method names. I opted to write an actual PNG file rather than PPM values as an exercise for myself, but shorter answers would definitely be possible if I used the PPM format.

I already start out as a loss by having to declare a variable to seed srand48 with time. I chose this over arc4random() or arc4random_uniform() because ultimately I would lose more bytes with those. I seed the rng to use drand48 to generate a random color and pick when to write a color to a pixel.

CGSize vs CGSizeMake and CGRect vs CGRectMake:

I switch between the inline C API functions and their Swift extensions to find the shortest constructor for each. CGSizeMake ends up being shorter than CGSize(), and CGRectends up being shorter than CGRectMake():

CGSizeMake(5,5)
CGSize(width:5,height:5)

CGRect(x:x,y:y,width:1,height:1)
CGRectMake(CGFloat(x),CGFloat(y),1,1)

I would have to create CGFloats x and y due to the nature of the for loop. I'm really not thrilled with the 2D loop and if equality checks, but I was really struggling to find a shorter way. There's definitely room to shave off a few bytes here.

Calling CGContextFillRects with an array of CGRect structs is cheaper than calling CGContextFillRect twice with two different values, so I save a few bytes with the array and pointer.

I also save 27 bytes by not calling UIGraphicsEndImageContext(). While this would normally be a "bug" in production code, it's not necessary for this toy program.

Colors:

Colors are also a pain to deal with, since I'm creating UIColor objects but need to write a CGColor opaque type to each pixel. The shortest code I found to create a random color was to use the UIColor constructor and get the CGColor from that UIColor. Same with white. If I were using the Cocoa framework instead of Cocoa Touch, I might be able to save some bytes using CGColorGetConstantColor(), but unfortunately that method is unavailable in the Cocoa Touch SDK.

Writing to to file:

Writing to a file takes almost 100 bytes. I'm not sure how to even go about optimizing this. On some systems depending on your permissions, you may need to use the Documents directory which would be even longer:

UIImagePNGRepresentation(i)!.writeToURL(NSFileManager.defaultManager().URLsForDirectory(.DocumentDirectory,inDomains:.UserDomainMask).first!.URLByAppendingPathComponent("a.png"), atomically:true)

Definitely open to further optimizations.

Edit 1: Saved a few bytes by rearranging some variable declarations.

Dyalog APL, 43 42 bytes

'P3',5 5 256,∊(3⍴240)(?3⍴256)[⌈(⊢+⌽)?5 5⍴0]

'P3',5 5a,∊(3⍴240)(?3⍴a←256)[⌈(⊢+⌽)?5 5⍴0]

Right to left:

?5 5⍴0 generates a 5×5 matrix of random numbers between 0 and 1 (but never 0 or 1)

(⊢+⌽) is a train that adds the matrix with its reflection

⌈ ceiling, returns 1 or 2 for each element

(3⍴240)(?3⍴256) colours - white-ish and a random one

[ ] use each element of the matrix as an index in colours

'P3',5 5 256,∊ flatten and add header

PHP, 236 bytes

I know this is an old challenge but I just like to challenge myself.

$o=imagecreate(5,5);$t=imagecolorallocate;$r=rand;for($c=[$t($o,240,240,240),$t($o,$r(0,255),$r(0,255),$r(0,255))];$y++<5;)for($x=-2,$a=[];$x<3;)imagesetpixel($o,$x+2,$y,isset($a[$v=abs($x++)])?$a[$v]:($a[$v]=$c[$r(0,1)]));imagepng($o);

Ungolfed:

// Store these frequently used functions
$t=imagecolorallocate;
$r=rand;

// Create 5x5 image
$o=imagecreate(5, 5);

// Define array of the possible colors
$c=[$t($o,240,240,240),$t($o,$r(0,255),$r(0,255),$r(0,255))];

// Loop over y axis
for($y=0;$y++<5;) {
    
    // This stores values for colors used in the current row indexed by the absolute value of x starting from -2
    $a = [];
    
    // Loop over x axis
    for($x=-2;$x<3;) {
        
        // Set a pixel of a random color for current coordinate. If it exists in the array, use the array value.
        imagesetpixel($o,$x+2,$y, isset($a[$v=abs($x++)]) ? $a[$v] : ($a[$v]=$c[rand(0,1)]) );
    }     
    
    // Empty the array
    $a = [];
}

// Output as PNG
imagepng($o);

Sample Output:

Javascript ES6, 296 bytes

Note: does not produce a file, draws to a canvas.

See it in action in this JS Fiddle.

s=20;W='#fff';M=Math;R=M.random;ctx=document.getElementById('c').getContext('2d');cr=`#${M.floor(R()*16777215).toString(16).slice(0,3)}`;f=Array(5).fill();a=f.map((_,i)=>f.map((_,j)=>R()*2|0));a.map((c,x)=>c.map((v,y)=>{ctx.fillStyle=y>=3?c[y==3?1:0]?cr:W:c[y]?cr:W;ctx.fillRect(y*s,x*s,s,s);}));

Uiua ^SBCS, 36 35 bytes

&ims≡⍉⍉⊂⇌↘1.⊏⁅⊞⋅⋅⚂⊚3⊚5⊟⚂_⚂_⚂▽3 0.94

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&ims≡⍉⍉⊂⇌↘1.⊏⁅⊞⋅⋅⚂⊚3⊚5⊟⚂_⚂_⚂▽3 0.94
                            ▽3 0.94  # gray color
                       ⚂_⚂_⚂         # random color
                      ⊟              # couple
             ⁅⊞⋅⋅⚂⊚3⊚5               # random 3x5 binary matrix
            ⊏                        # select
        ⇌↘1.                         # duplicate, drop first row, reverse
       ⊂                             # join
    ≡⍉⍉                              # rotate image
&ims                                 # display image

Ruby, 132 bytes

Outputs a PPM image.

c=(1..3).map{rand 256}
puts"P3 5 5 255",(1..15).map{rand 2}.join.gsub(/(..)./){$&+$1.reverse}.gsub(/./){($&==?0?c:["240"]*3)*" "+$/}

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