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.