This is extremely dependent on the personality of the betta and the tank size. Unless you have 10 gallons or more, you can really only add a snail or ghost shrimp, and even then you should have at the absolute very least 3 gallons, though 5 gallons would be better as snails have a pretty heavy bioload and shrimp need a lot of places to hide.
Most of these need similar care requirements as a betta, so a heater and clean water are paramount, though for ghost shrimp the parameters in which they can survive are even more narrow. Basically, they need cleaner water.
Have you considered plants? A Marimo moss ball is not only beneficial to your bioload but bettas often use them as lounge chairs.