This is a little bit of a mystery and my answer is a collection of possibilities that I've looked at. As you noted in your comment, he is still healthy and active.
In Discus and some Angel fish, there is a condition called 'Peppering' where the animal attempts to camoflague itself better by 'peppering' its body with black dots. This normally happens if the substrate (or tank generally) is dark. In your picture, it appears that you do have a fairly dark substrate but it's unclear whether this condition is caused only by the 'Pigeon Blood' strain of Discus or not.
Bettas are pretty popular and some fish may not be as pure-bred as others. Many enthusiasts have noted that their blue bettas have developed red fins over time. This still seems unlikely that they would be such finely defined spots.
General Water Quality
Check and recheck your water parameters to ensure that you have no ammonia, nitrite and a low level of nitrate (< 40ppm). Water quality can impact what a fish looks like but again, unlikely that it would cause black spots. This doesn't look like fin rot (which is normally indicated by ripped or torn fins exposing the vertebrae).
Monitor the small dots and make sure they aren't increasing in number, changing in size or moving position. Ensure that your water parameters are stable and you're doing all required maintenance.
If there are any changes, you could update your question.