The variable $id isn't in the scope of the function. You need to use the use clause to make external variables accessible:

$foo = array_filter($bar, function(id) {
    if (isset($obj->foo)) {
        var_dump(obj->foo == $id) return true;
    }
    return false;
});

See documentation for Anonymous functions (Example #3 "Inheriting variables from the parent scope").

The scope of the variables inside an anonymous function is ONLY within the anonymous function.

You need to inherit the variable from the parent scope. You can find more details about it in the PHP Documentation about anonymous functions (Example #3)

which would transform this line:

$filtered = array_filter($json, function (q) {

into this:

$filtered = array_filter($json, function (q, $val) {

Your $leagueKey variable is outside the scope of the anonymous function (closure). Luckily, PHP provides a very simple way to bring variables into scope - the use keyword. Try:

$leagueKey = 'NFL';
$response['response'] = array_filter($response['response'], function($tier) use ($leagueKey) {
    return $tier['LeagueKey'] === $leagueKey;
});

This is just telling your anonymous function to "use" the $leagueKey variable from the current scope.

Edit

Exciting PHP 7.4 update - we can now use "short closures" to write functions that don't have their own scope. The example can now be written like this (in PHP 7.4):

$response['response'] = array_filter(
    $response['response'], 
    fn($tier) => $tier['LeagueKey'] === $leagueKey
);

You have at least two options:

1. Globalize the desired variable, and then reference it inside the callback
2. Wrap the score calculation logic with a function, then reference it inside the callback

Globalize the Variable

<?php
global $score;
$score = 42; //Some crazy calculation I don't want to repeat.

function add_score_to_title($title) {
    global $score;
    return 'Quiz Results (' . $score . '/') - ' . $title;
}

add_filter( 'aioseop_title_single', 'add_score_to_title');
?>

Wrap the Score Calculation

If you only ever need the score calculation inside the filter, pull the logic into the callback itself:

<?php
function add_score_to_title($title) {
    $score = 0;
    $questions = get_quiz_result_questions();
    $total_questions = 0;
    foreach( $questions as $question ) {
        $order = $question->order;

        if( $order >= 100 ) {
            break;
    }

    if( $question->correct == $_POST['Q'][$order] ) {
        $score++;
    }
    $total_questions++;

    return 'Quiz Results (' . $score . '/') - ' . $title;
}

add_filter( 'aioseop_title_single', 'add_score_to_title');
?>

Better yet, you could wrap your score calculation in a function of its own, and then call that function inside your callback:

<?php
function wpse48677_get_score() {
    $score = 0;
    $questions = get_quiz_result_questions();
    $total_questions = 0;
    foreach( $questions as $question ) {
    $order = $question->order;

    if( $order >= 100 ) {
        break;
    }

    if( $question->correct == $_POST['Q'][$order] ) {
        $score++;
    }
    $total_questions++;
    $output['score'] = $score;
    $output['total_questions'] = $total_questions;

    return $output;
}

function add_score_to_title($title) {

    $score_results = wpse48677_get_score();

    $score = $score_results['score'];

    return 'Quiz Results (' . $score . '/') - ' . $title;
}

add_filter( 'aioseop_title_single', 'add_score_to_title');
?>

If you find you have problems referencing the $_POST object, you can also register your query variable and then use get_query_var() internally to get data:

function add_score_query_vars( $query_vars ) {
    $query_vars[] = 'Q';

    return $query_vars;
}
add_filter( 'query_vars', 'add_score_query_vars' );

With this in place, $_POST['Q'] can be replaced with get_query_var('Q').

Let's use array_map() to explain what's going on.

We want to duplicate the input of an array: so if the input is aa, the output would be aaaa.

So the normal way, would be to create a function and then pass it to array_map():

$array = range('a', 'e');

$new_array = array_map('duplicate', $array);
print_r($new_array);

function duplicate($string){
    return $string.$string;
}

Online demo

But what if you want to use this function only once ? Since PHP 5.3, there is something called anonymous functions, we use it like the following:

$array = range('a', 'e');

$new_array = array_map(function($string){
    return $string.$string;
}, $array);
print_r($new_array);

Online demo

Now, let's say for example you want to add a standard value from another variable. That's easy with global variables. But as we know, global variables are evil and should be avoided. We may use use():

$array = range('a', 'e');
$standard_value = ',';

$new_array = array_map(function($string)use($standard_value){
        // $standard_value becomes available inside the function
    return $string.$standard_value.$string;
}, $array);
print_r($new_array);

Online demo

use() can be become also useful if we use a reference to write to an external variable while looping:

$array = range('a', 'e');
$another_string = '';

$new_array = array_map(function($string)use(&$another_string){// note &
    $another_string .= $string.$string; // overwrite $another_string
    return $string.$string;
}, $array);
print_r($new_array);
echo PHP_EOL . $another_string;

Online demo