My 15 year old son is into biking and he says, "for an 11 year-old, about the top of the line would be a Haro BMX. It's just a good all-around strong bike for riding and jumping and dirt riding. The cost is around $250. But if you're not looking to spend that much, Target or Walmart carry cheaper BMX bikes that just aren't as well-built, but still fine for dirt riding and general use."
He says you don't need anything "fancy or expensive" unless he's planning on getting involved in the sport.
Good luck and have fun!