There are a few Invertebrate options. No fish though as most fish dont handle a number of factors that a 3 gallon might encounter of regularly does, such as aggression from the betta, ammonia reaching high, toxic levels, constant stress due to netting, along with moving out of the tank, and a space issue, most freshwater fish are far too active to be placed in that small of a tank resulting in them not showing there full potential. Actually it is highly likely that the betta will become nippy, and start tailbiing as most fish will be seen as zooming around in that tank to the betta due to how small it is.
most of the options below, I have done with success (ghost shrimp were hard to keep alive, though) which is why I recommended them.
-Pond Snails, These would be a great choice, as they have very minimal requirements, along with a very small bio load due t how small they are. you dont have to feed them, no special care. and they wont have a population boom as they do in community tanks unless you feed flakes, or your betta doesnt eat all the pellets. they also will provide as great cleaner crews in respect to decaying plant matter. Please take into consideration that these are known to over populate the tank when given enough food, ideally you would aim for a maximum of around 5
-Amano Shrimp, much hardier than the other shrimp and will tolerate 100% water changes alot better, along with not perfect water quality. They are very entertaining and fun to keep, Amano's will thrive with planted tank along with added supplements like shrimp pellets. I would highly recommend these, very awesome shrimp
-Ghost Shrimp, not much these as they require pristine water quality along with a minimal amount of stress. with 100% weekly water changes it will be very stressful
-Cherry Shrimp, not much these as they require pristine water quality along with a minimal amount of stress. with 100% weekly water changes it will be very stressful
Your best choices would be amano shrimp.