Here's my $0.02:
There are a bunch of ghey webmasters here giving you advice about a situation they know nearly nothing about. We don't know what happened between you two, how things ended, and what the actual status of things between you two is right now. If you haven't had any contact in 3 years and just send her a gift out of the blue that shows you spent a lot of time and effort picking it out along with a really long and emotional card then that comes off as some creepy stalker shit.
If you want to reconnect just send her a message on Facebook (because this would be the least amount of effort and you would have known it was her B-day because FB tells you these things...not because you use you tears as lube and go through a box of Kleenex every time her b-day rolls around) or give her a call on her birthday to wish her well.
If you really want to send her a little gift, send her something that was part of an inside joke that is inexpensive and will remind her of the good times you had together and what a great fucking guy you are/were. (Btw, if you cant think of anything like this off-hand, chances are the relationship isn't worth rekindling). Send that off with a little note about how you saw it and it made you think of here and remember her birthday...yada yada yada.
In the end you have to make a judgement call for yourself because only you know the details of the situation. Sacking up and pursuing what you want in life is alpha. Disregarding your feelings for this girl (if you had something meaningful and have reason to think she would be interested in seeing you again) because some ghey webmasters you dont know tell you to is beta. Failing to act one way or another and not moving forward with your life is fucking omega.
Best of luck with the situation. In the end it will all work out like it should whether you get back together with her or find out from personal experience that her best friend loves to be punished in the ass... :thumbsup: