rent a boat in the Greek islands

Your question is not specific enough. Every Greek island where there's a chance to make money with a boat rental business will have at least one. Some islands have several. Some will only rent you a boat with a driver, some let you do the driving. Focus your search on those islands that interest you, and do a Google search for rental companies there, using keywords such as "Naxos Boat Rentals".

I have hired a small boat for the odd day several times in Greece. Do you mean this or chartering a larger yacht or catamaran for a week? I have been on several flotilla and bare boat holidays in Greece through Sunsail. Are you a qualified skipper? You will need Day Skipper qualifications to take a boat on your own.

Sorry for the lack of info...so we were thinking we might rent for a day, with a driver. We will be in Mykonos for a day in late September.