Hi.
You seem very clear about wanting a closer connection with your mom, and you're certainly entitled to try and have one! Many people raised by one parent have similar yearnings at some point as they get older.
I asked Dr. Frederyka Shabry, a psychiatrist who works with teens, for tips on where you might start. She suggests exploring new ways of reaching out to your mother and also educating yourself about any emotional or legal reasons why you've never seen each other very much.
Do you have siblings, grandparents or other relatives or friends who might know the history? If you're going to try and change things between you and your mom, you probably want to find out how they got the way they are now. Having the facts could also help keep you from taking it personally if your mom doesn't react as well as you'd like to your efforts.
As for reaching out, maybe send a birthday card or start phoning (or emailing) more than you might be already. If your mother doesn't reciprocate, it's possible that she's not ready for a more involved relationship with you--at this particular time in her life. Of course, you can always try again later...
Without knowing more about your situation, I'd hesitate to suggest more specific strategies. I would, however, recommend that you speak to a counselor of some kind, maybe at school, or check out agencies and websites that deal with family issues. Online discussions such as iconnect could be a good place to start.