Instead of opening php page via javascript, you need to change the form action dynamically to point to your generated url.

HTML:

<form name="login">
    Username: <input type="text" name="user_id"/>
    Password: <input type="password" name="pswrd"/>
    <input type="submit" onclick="check(this.form)" value="Login"/>
</form>

JavaScript: (line 9 & 10 changed)

function check(form) {
    var userCredentials = [["jsmith", "smithpassword", "John Smith"], ["jdoe", "doepassword", "Jane Doe"]];
    var credCheck = 0;
    for (var i = 0; i < userCredentials.length; i++) {
        if (userCredentials[i][0] == form.user_id.value) {
            credCheck += 1;
            var displayName = userCredentials[i][2];
            if (userCredentials[i][1] == form.pswrd.value) {
                form.action = "home.php?display_name=" + displayName;
                return true;
            } else {
                alert('The username and password do not match.');
                return false;
            }
        }
    }
    if (credCheck == 0) {
        alert('The username entered is not valid.');
        return false;
    } else {
        return true;
    }
}