We had a great time in Phuket staying at BYD Lofts (1 bedroom serviced apartment). It's not the cheapest place to stay, but it was certainly worth it and we will definitely stay there again. Rooms spacious, clean and well outfitted; all the staff we met were friendly and very helpful; and the service was excellent at all times. It's close enough to the centre of town, but far enough way to make it a bit quieter and a nice haven, especially with our young son along.
Some small ideas for an even better experience - get at least one more high chair for the restaurant, maybe expand the breakfast selection even more (more options for cooked items to order?)