I just looked on Petfinder.com and found a rescue organization that may be near you. Many of the dogs would not be right for your allergic 3 year old, but you might want to check out their website.
The rescue's name is Shih Tzus & Furbabies Small Breed Dog Rescue. The website is http://www.stfbr.org/
A notice on their available dogs page announced a Meet and Greet in June in Bell Meade, NJ. I don't know if that is close to you or not. I am not very familiar with New Jersey.
They have Havanese mixes listed on their site, but I don't see any purebreds right now. One of them is a HavaMalt which might work for allergies, but you need to be very careful with a small dog around very young children. It often is not a good mix. sometimes getting an adult dog, although small, may be a better situation than a puppy.
Another example: There is a pair of Cotons [which might be ok for an allergic person] on the site that need to be adopted together, but they are nervous around small children
Anyway, you might want to look on Petfinder.com often because things change quickly.
Good luck with your search.