Remove duplicate array elements based on part of their content?

It would be helpful to know the context of your problem.

This function returns all strings with unique href value, but does nothing about managing closing tags. Removing closing tags would be a complex task. Plus I'm pretty sure parsing HTML with regex is not a good idea.

function sortByHref (array) {
  var hrefReg = new RegExp('href="(.*)"');
  var seen = {};
  var match, href;
  return array.filter(function (x) {
    match = hrefReg.exec(x);
    if (match) {
      href = match[1];
      if (seen.hasOwnProperty(href) && seen[href]) return false;
      seen[href] = true;
    }
    return true;
  });
}

There has to be another way to solve your problem, if you have described what exactly are you trying to accomplish.

Here is a purposely verbose solution for an easier understanding. I am assuming that tags without a href value will simply remove duplicates based on whole string.

var arr = [
    '<tag href="cheese.html">',
    '<tag href="cheddar.html"></tag>',
    '  <tag href="cheese.html"></tag>',
    '</tag>',
    '<tag href="burger.html">',
    ' <tag href="burger.html">',
    '   <tag href="burger.html"></tag>',
    ' </tag>',
    '</tag>'
];

// Remove whitespaces on both ends from each string in array
// Not a necessary step, but will just handle leading and trailing whitespaces this way for convenience
arr = arr.map(function(tagString) {
    return tagString.trim(); 
}); 

// Regex to retrieve href value from tags
var hrefRegexp = /(\s+href=\")([^\"]+)(\")/g;

// Create an array with just the href values for easier lookup
hrefArr = arr.map(function(tagString) {
    // Run regex against the tag string
    var href = hrefRegexp.exec(tagString); 

    // Reset `RegExp`'s index
    hrefRegexp.lastIndex = 0; 

    // If no href match is found, return null, 
    if (href === null) return null; 

    // Otherwise, return the href value
    else return href[2]; 
});

// Store array length (this value will be used in the for loop below)
var arrLength = arr.length; 

// Begin from the left and compare values on the right
for (var leftCompareIndex = 0; leftCompareIndex < arrLength; leftCompareIndex++) {
    for (var rightCompareIndex = leftCompareIndex + 1; rightCompareIndex < arrLength; rightCompareIndex++) {

        // A flag variable to indicate whether the value on the right is a duplicate
        var isRightValueDuplicate = false; 

        // If href value doesn't exist, simply compare whole string
        if (hrefArr[leftCompareIndex] === null) {
            if (arr[leftCompareIndex] === arr[rightCompareIndex]) {
                isRightValueDuplicate = true; 
            }
        }

        // If href value does exist, compare the href values
        else {
            if (hrefArr[leftCompareIndex] === hrefArr[rightCompareIndex]) {
                isRightValueDuplicate = true; 
            }
        }

        // Check flag and remove duplicate element from both original array and href values array
        if (isRightValueDuplicate === true) {
            arr.splice(rightCompareIndex, 1); 
            hrefArr.splice(rightCompareIndex, 1); 
            arrLength--; 
            rightCompareIndex--; 
        }
    }
}

console.log(arr); 

/* Should output
[ '<tag href="cheese.html">',
  '<tag href="cheddar.html"></tag>',
  '</tag>',
  '<tag href="burger.html">' ]
  */