I have a good bit of experience with thos sort of thing. The suggestion below to shatter the edges is a good one. You often times must make it look worse before making it look better. If the hole is uneven and jagged to begin with you are fine, otherwise you need to make it look like a natural "pock mark" and deliniate the shape of the hole. I disagree with using an acrylic fill as this will loosen up over time and even though it would be colored it would tend to bleach out and whiten over time while exposed to water. The best solution is to use a 2=part penetrating epoxy, such as TouchStone from Bonstone. Repair kits are available in small quantity from stone tool suppliers and can be colored to match teh other fill in the travertine. Above all make sure to prep the area with acetone first and let it dry well before using the epoxy. After filling and smoothing you can go through the refinish process with polishing discs on a hand tool.
this is my first posting/ question so be gentle with me. i have a manufacturer of travertine sinks who sold a sink to someone in the Philly area and the sink has a hole in it ( .25 - .5 inch big ). she is trying to avoid having the sink deinstalled and reinstalling a new one and wants me or someone to fix it.i need to know if anyone knows about a travertine hole filling material that will work on this sink (must hold water). i am open to any/ all suggestions. thanks.......barry