Neon tetras can be rather sensitive to a lot of conditions in the first place, unrelated to living with the Betta fish. They could have died for a number of reasons, and they can be a bit tricky to keep. Somebody suggested cories, and those should be fine with your Betta. As far as critters go, you could also put a couple of ghost shrimp in there. Even if your Betta does eat the shrimp, they're like 2 bucks for a bag of ten so It's a fairly inexpensive addition, and you won't be losing too much money. Just be careful not to overcrowd the tank, as always.
Best of luck to you.