I know what you're talking about. I don't think its anything to worry about, i've seen a few Bettas with them on their little pectoral fins, i'll look and see if I can find a photo of another Betta with it as well.
As far as I know it's nothing to worry about.
Here is a photo of one of my boys.
They look similar to the right pectoral fin, correct? I've only ever seen them on Crowntails, it seems like it's nothing to worry about.
Also you should change his water twice a week since he's in a 1 gallon tank. A filter for a tank that size is not needed since ammonia can build up quicker then the filter will be able to clean the tank water. You should be doing a 50% water change then a 100% water change twice a week to keep him healthy.