There are a lot of possible places that might work for you, so it’s hard to narrow my suggestions down without knowing where you are starting from and how far you are willing to travel from there. But two things that come to mind are Bali, Indonesia, and the Cancun area of Mexico. If you are in the US or Canada then it will obviously be a long flight to Bali, but things are quite cheap once you get there so it’s really worthwhile. The island has beaches and water sports, but also really interesting culture, not to mention loads of shopping and beach resorts. Its main problem these days is vehicle traffic, so getting around the island is slower than it used to be.
Brazil, on the other hand, has a famous problem with petty crime in tourist areas so I wouldn’t recommend it for a woman and young child. Another to consider is Puerto Rico, which has a lot to offer including great weather, and it’s also more or less part of the US, so it’s fairly safe and well organized. Hotels in Puerto Rico are a bit expensive, but you should be able to get an apartment rental in the San Juan area (gorgeous, by the way) for a modest price if you are staying for more than a few days. Let me know if you have any other questions. -Roger