Oh yes, you can train your betta, sounds soo cute, I have my bettas trained to jump for food, by putting a pellet on my finger, I have one of my bettas sing..lol..(you know when they move their mouth really fast, like they are yelling at you) well,I did that with one of my fish, while playing music..it is hysterical..and yes, they know their names if you call them by their names a lot, and they will recognize your voice, as well.
Bettas are very intelligent, and they have the ability to be trained like dogs. I talk to my bettas all the time, spend a lot of time watching, and observing all their activity..they are a lot of fun, and wonderful lil fishy's to have.
There is a Betta Training Kit, that I saw on Amazon.com and you can train your betta to play with a ping pong ball, and swim in an out of hoops, it's a very cool kit.