The following snippet shows how to replace the user display name by a custom field. Just uncomment the replacement you want to keep.

If you want to show a custom user meta value, use the WordPress get_user_meta( $user_id, $key, $single ) function

function my_prefix_replace_display_name_on_leaderboards( $output, $leaderboard_id, $position, $item, $column_name, $leaderboard_table ) {
    $user = get_userdata( $item['user_id'] );
    // Replace the user display name by the user nickname (alias), eg: jhondoe
    //$output = $user->nickname;
    // Replace the user display name by the user email, eg: jhondoe@gamipress.com
    //$output = $user->user_email;
    // Replace the user display name by the user login (not recommended), eg: jhondoe90
    //$output = $user->user_login;
    // Replace the user display name by the first name, eg: Jhon
    //$output = $user->first_name;
    // Replace the user display name by the last name, eg: Doe
    //$output = $user->last_name;
    return $output;
}
add_filter( 'gamipress_leaderboards_leaderboard_column_display_name', 'my_prefix_replace_display_name_on_leaderboards', 10, 6 );