How can I repair a hole in bathroom wall tile?
You've got three separate problems:


You have a hole in the tile 
You have a hole in the sheetrock/backer board behind the tile
Water may have damaged sheetrock behind other tiles.


I'd start by removing the grout around the broken tile with a grout saw, and prying the tile off.

Next, I'd take a look at the rest of the wall behind the tile... if it is in good condition, I'd make a sheetrock patch using cement backer board.
Next I'd try to find a tile that looks close enough and replace the broken tile with it.
Alternatively, you could cut the broken tile square (so it looks like the US state of Utah), and find a complementary color and cut that to fit the missing piece.

Re-install the missing tiles and re-grout the area.

If the wall has a lot of moisture,  I'd stick a dehumidifier in there for a while and see if the wall can be saved. Otherwise this could get quite expensive quite quickly...