When a chip is reballed, the solder pads are cleaned off and new solder is added. The newer chips don't have wires that extend down to the board like the older stuff. They have pads. They are spread out along the underside of the chip. These don't tolerate heat cycling as well and the solder joint can crack. There are two basic ways to fix it. Reflow the solder by putting a little bit of liquid flux under the chip, letting it wick under, then heating up the chip in a prescribed manner that will cause the solder to melt and rebond. This is relatively cheap and easy to do but because of the solder type it can recrack again at a later time. The better method is to heat the chip and pull it off. Then clean all the solder and put a newer solder that has better crack resistance on each of the pads. This is the "reball" as the solder will sit on each pad like little balls. Then you place the chip back down on the board and do the prescribed heating to melt the solder and have it bond to the contacts on the board.
If they reballed it then that's good. And also explains the cost. I've only done the reflow method and it works but you have to keep in mind the risk. Reballed done right with the correct solder should be better then the factory. Also looks like they installed new memory too.
Don't know what a BGA chip is. Will have to look that up
Edit: just looked up BGA. It's just "ball grid array". Basically the arrangement of the pads or contact points on the chip.
Older style chips
Newer style with the ball grid
And the solder pads underneath
Here's a good picture of the "balls" of solder