Expand string without Invoke-Expression

Embedding variables into strings is not the only way to create dynamic text, the way I would do it is like this:

$WelcomeMessage = 'Hello {0}, today is {1}'

# 100 lines of other functions and what not...

function Greet-User
{
    $Username = Get-UserNameFromSomewhereFancy
    $Date = Get-DateFromSomewhereFancy

    $WelcomeMessage -f $Username, $Date
}

The canonical way to delay evaluation of expressions/variables in strings is to define them as single-quoted strings and use $ExecutionContext.InvokeCommand.ExpandString() later on.

Demonstration:

PS C:\> $s = '$env:COMPUTERNAME'
PS C:\> $s
$env:COMPUTERNAME
PS C:\> $ExecutionContext.InvokeCommand.ExpandString($s)
FOO

Applied to your sample code:

$WelcomeMessage = 'Hello $UserName, today is $($Date.DayOfWeek)'

...
...
...

function Greet-User {
  $Username = Get-UserNameFromSomewhereFancy
  $Date = Get-DateFromSomewhereFancy

  $ExecutionContext.InvokeCommand.ExpandString($WelcomeMessage)
}

Have you considered using a lambda expression; i.e. instead of defining the variable as a string value define it as a function, then invoke that function passing the relevant parameters at runtime.

$WelcomeMessage = {param($UserName,$Date);"Hello $UserName, today is $($Date.DayOfWeek) $([void](remove-item c:\test\test.txt))"}

#...
# 100 lines of other functions and what not...
#...

"testfile" >> c:\test\test.txt #ensure we have a test file to be deleted

function Get-UserNameFromSomewhereFancy(){return "myUsername";}
function Get-DateFromSomewhereFancy(){return (get-date);}

function Greet-User
{
    $Username = Get-UserNameFromSomewhereFancy
    $Date = Get-DateFromSomewhereFancy

    $WelcomeMessage.invoke($username,$date)
}

cls
Greet-User

Update

If you only wish to allow variable replacement the below code would do the trick; but this fails to do more advanced functions (e.g. .DayOfWeek)

$WelcomeMessage = 'Hello $Username, today is $($Date.DayOfWeek) $([void](remove-item c:\test\test.txt))'
#...
# 100 lines of other functions and what not...
#...

"testfile" >> c:\test\test.txt #ensure we have a test file to be deleted

function Get-UserNameFromSomewhereFancy(){return "myUsername";}
function Get-DateFromSomewhereFancy(){return (get-date);}
function Resolve-WelcomeMessage(){
    write-output {param($UserName,$Date);"$WelcomeMessage";}
}
function Greet-User
{
    $Username = Get-UserNameFromSomewhereFancy
    $Date = Get-DateFromSomewhereFancy
    $temp = $WelcomeMessage 
    get-variable | ?{@('$','?','^') -notcontains $_.Name} | sort name -Descending | %{
        $temp  = $temp -replace ("\`${0}" -f $_.name),$_.value
    }
    $temp 
}

cls
Greet-User

Update

To avoid code injection this makes use of -whatif; that will only help where the injected code supports the whatif functionality, but hopefully better than nothing...

Also the code now doesn't require parameters to be declared; but just takes those variables which are available at the time of execution.

$WelcomeMessage = {"Hello $Username, today is $($Date.DayOfWeek) $([void](remove-item c:\test\test.txt))"}

#...
# 100 lines of other functions and what not...
#...

function Get-UserNameFromSomewhereFancy(){return "myUsername";}
function Get-DateFromSomewhereFancy(){return (get-date);}
function Resolve-WelcomeMessage(){
    write-output {param($UserName,$Date);"$WelcomeMessage";}
}

"testfile" >> c:\test\test.txt #ensure we have a test file to be deleted

function Greet-User {
    [cmdletbinding(SupportsShouldProcess=$True)]
    param()
    begin {$original = $WhatIfPreference; $WhatIfPreference = $true;}
    process {
        $Username = Get-UserNameFromSomewhereFancy
        $Date = Get-DateFromSomewhereFancy
        & $WelcomeMessage 
    }
    end {$WhatIfPreference = $original;}
}

cls
Greet-User