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• jQuery is entirely unnecessary and will just make your solution slower
• Don't initialize to new Class() - this is not Java. Instead, initialize to the initial value, or undefined (var x;).
• use a function to encapsulate your verification logic, and to separate it from the DOM interface.
• don't call me.val() more than once; cache the result
• instead of the slow $.each, use a simpler loop
• You already know the length of the id string, so you can get rid of the p%2 check and simply operate twice on each loop, once on the current position and once on p + 1, and increment by 2 each time (this also results in half as many iterations, speeding up the loop even further).
• Cast using +string (resulting in a number) and '' + number (resulting in a string)
• split the string into the first 12 and the last 1 at the beginning, so you don't have to keep excluding the last character and re-extracting it.
• 'string'.slice(-1) is a more concise way to access the last character in a string
• check to make sure that not only is the string 13 characters long, but also that every character is a numeral

With all that and a few stylistic preferences, here's my take on your problem:

function isValidSAID(id) {
    var i, c,
        even = '',
        sum = 0,
        check = id.slice(-1);

    if (id.length != 13 || id.match(/\D/)) {
        return false;
    }
    id = id.substr(0, id.length - 1);
    for (i = 0; c = id.charAt(i); i += 2) {
        sum += +c;
        even += id.charAt(i + 1);
    }
    even = '' + even * 2;
    for (i = 0; c = even.charAt(i); i++) {
        sum += +c;
    }
    sum = 10 - ('' + sum).charAt(1);
    return ('' + sum).slice(-1) == check;
}

if (!isValidSAID(document.getElementById('id').innerHTML)) {
    alert('Your South African ID is not valid.');
}