A tile dealer will sell a bit that cuts holes in tile to go around pipes for tub spouts and shower heads. They are similar in structure to the 'hole saws' that cut out a hole for a door knob. I have a small one like you need for spouts and a larger one for faucet valves. These diamond (or cheaper carbide) drill bits are expensive- about $50 as I remember, but you could get a cheaper carbide tipped one for your single purpose. These cheaper one's have tiny carbide bits welded onto the cutting surface.
FORGET any kind of large carbide drill bit (the type that are shaped like a typical drill bit) as they won't make a clean hole and will risk breaking/chipping the tile.
If your hole does not have to be clean- that is there is a plate around that button that will hide considerable error in cutting, you could buy a glass bit for about $8 at a hardware store and drill several holes in a circle, getting your approx. 1" hole.
BTW you'll need a larger drill for the hole saws.
thetiler