Despite the rain that frequently falls around Braulio Carillo, birding here can be very good. There are two birds that can be observed here that are more easily seen than other places... Black-and-Yellow Tanager and Lattice-tailed Trogon. There is also an old tapir reserve known as "El Tapir" which now serves as a hummingbird garden. This may be worth a stop as you can sometimes pick up Snowcap as well as 5-6 other foothill hummingbirds.
There are no good places to stay close by this park, but San Jose and the La Selva area are both within a 40 minute drive.
For more information, on this park you might check out Pat O'Donnells the "Costa Rica Living and Birding" blog. Pat might also be available for one-day guide service around Braulio Carillo.