Weather on the Kona side of the island should be fine, like it is pretty much year-round. There is more rain as we move into winter, temperatures are slightly cooler, but dive conditions are generally good in Kona all year. Now, the Hilo side (where I live) is different. We are exposed to the trade winds, and winter typically brings cloudy days, rain, and rougher water. Except when it doesn't
There can be days with very nice weather in Hilo and good diving in the winter, but Kona is always a MUCH safer bet this time of year.
Visibility is generally very good and water temp warm in Kona year round. There is a chance of rougher water conditions in the winter, but November should be fine. If you have not gone diving in Hawaii, the water is a little colder than many people expect. Generally, a 3mm full is good for most folks, but a 5mm full is not "too much" in the winter if making multiple dives over several days. It all depends on your personal tolerance.
As far as location centrally-located for diving, I'd recommend a hotel in Kailua-Kona. You'll want to rent a car (the Big Island is BIG). There are a number of good dive ops. I typically shore dive, but have gone out with Jack's Diving Locker and liked them a lot.
The night Manta Ray Dive is a must-do dive while in Kona. Most of the dive ops do it. Our family did it with Jack's and had a great time.
I do not have a specific hotel recommendation. My feeling over the years is that several of the hotels in Kailua-Kona that were once really nice have slipped a lot in quality. I'll still stay at them when I go to Kona for the weekend, but have trouble recommending them to others as a "vacation" destination. I think the nicer hotels are now on the Kohala coast, but they are about 20-30 minutes out of Kailua-Kona, so not nearly as convenient for diving. Hopefully one of the Kona divers or ops will be able to give a good lodging recommendation.
Anyway, good luck, and enjoy your trip.