I want to display the users birthday when they log in next to where it shows their username and the option to sign out.

We provide custom modifications for a fee only.

Whatever you need to code for this, is to catch the value of user's birthday in userLoggedIn function, and assign it to some global variable, which may be pasted to user_logged_in template afterall.

How much would that be?

Just this function, if we consider the total time, could be implemented regarding our custom rates. Count not less than a hour in total for all that, i.e. $40.

However if you'd need some other custom codes for your forum, the overall could be less (since for more tasks, we don't need to investigate what's on your forum each time, spend time to connections etc.).