How to get value of nested attributes of element

Following is commented to show steps

// target element
var $picker = $('domain-picker'),
 // parse current attribute value string to object
current= JSON.parse( $picker.attr('current'));
// change value
current.value ='someOtherString';
// stringify object and put back as attribute value
$picker.attr('current', JSON.stringify(current));

Modify your HTML, Use single quotes for current attribute

<domain-picker class="pull-right" 
current='{"label":"AMN/GRP","value":"assf2324234"}' in-header="true" show-in-header="true">
</domain-picker>

in jQuery

var json = $("domain-picker").attr("current");
json = JSON.parse(json)
var value = json.value;

Here is working demo https://jsfiddle.net/758y0fp1/1/

As @charlietfl mentioned, if you wanna use jQuery, his answer is the 1 you need, this can be done with pure Javascript too, ie.

// getting the element
var picker = document.getElementsByTagName('domain-picker')[0];
// parsing current attribute string to object
var current= JSON.parse(picker.getAttribute('current'));
// changing the value
current.value ='newString';
// stringify the object and set the attribute again like
picker.setAttribute('current', JSON.stringify(current));

You need to first get the value of attribute .attr() in jquery or getAttribute() in javascript will help. Now since the value is an invalid JSON(as per the html provided by you), convert the invalid JSON to valid and then select the value.

var obj = document.querySelector("domain-picker").getAttribute("current");
var json = obj.replace((/'/g), "\"");
var obj = JSON.parse(json)

//console log the retrieved value.
console.log("value :", obj["value"])

// set attribute now

obj["value"] = "1234567";


document.querySelector("domain-picker").setAttribute('current', JSON.stringify(obj));

<domain-picker class="pull-right" 
current="{'label':'AMN/GRP','value':'assf2324234'}" in-header="true" show-in-header="true">
  Hello inpsect me to see the changed attribute value.
    </domain-picker>