- 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.');
    }