It would be highly unusual if you didn't experience heavy rains. The rainy season is from mid-to-late September through late November. You will be there in the prime rainy season. You will wonder if it is ever going to rain. You will feel as if nothing ever gets dries. You are in the Bay Islands - get over it and greet each new, albeit rainy day with excitement and contentment, knowing that you are in the Bay Islands.
Some people will suggest that the no-see-ums are less of a problem during the rainy season. This is mostly wishful thinking. No-see-ums are simply a given in the Bay Islands. The only real issue is how sensitive you will be. Some people will tell you how they never got bite or just got bite once (i.e., see below). People vary in the degree to which bugs find them (CO2 is thought to be attractant, so something as simple as breathing patterns could be an issue) and the degree to which they respond to bug bites (little, if any response to major welts). When in doubt, assume that you will have many bites and will itch like crazy. Therefore, brings lots to DEET-based products and, if you are medically ok for such things, OTC products such as Benadryl and Calamine lotion. If you are one of the lucky folks that doesn't react, thank your lucky stars and pass the products on to friends who didn't bring enough.
Malaria? Locals approach malaria in terms of not "if I get it", but "when I do get it." And it is nasty. Some people will do drugs like chloroquine on a prophylactic basis. Speak to your doctor re what's right for you.
Yes, the Bay Islands do have their drawbacks, but there are few places that can compare with the positive features of these islands! Enjoy and let us know how you are doing.