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.
Remember that the tank your fish lives in is his or her's whole world. Ask yourself this question, and answer it honestly: would you choose to live there?