I love Thanksgiving week and also early to mid December in Key West. The island is decorated for Christmas and there are lots of special events (lighted boat parade, Christmas parade, Christmas lights trolley tours, etc.) in December without the crowds of Christmas week. It can get a little chilly at night (for this Floridian) but the weather is mostly gorgeous.
My husband prefers the weather in the summer (hot and very humid) but it can be miserable if you aren't used to it. It's not bad if you plan to spend a lot of time in the water during the day. Early May is a nice time to visit - still hot but not quite as humid. Songwriters Festival is in early May and a very popular time to visit but we have been able to get that week at Sunset Harbor by requesting it very early.