If you really feel you must try to get back together, then instead just send her a message like "hey, isn't today your birthday? i know it's either yours or some other girl.."
Duder,
I'm a dating and attraction coach in my spare time, so take it from someone who knows... No dude no.
If you must, moxie had pretty good advice. Better make it something stupid and funny, not serious and meaningful or You'll just look like a pathetic wussbag.
I recently sent an out of state friend of mine a stick horse for her birthday, no explanation, no warning. She friggin loved it.
If you want to take it further, pm me.
If you give her something out of the blue on her birthday after 3 years, it's gonna look really creepy. If you really want to get back with her, you need to just call her up, all chill and shit, and make it sound like you just randomly thought of her.
Say something like, "Hey I saw something random that reminded me of you the other day and remembered we haven't chilled in a while. Wanna catch up over a coffee or something?"
If you let her know how much you've really been thinking of her, you'll have no shot.