I'm not going to bash you out for getting a new betta because I believe you can cycle a fish in a tank AS LONG AS you know what you're doing. You need a test kit, and to moniter the ammonia, nitrates and nitrites levels and do water changes accordingly.
What I think you should do, because you haven't had your water tested, is maybe put him in a gallon of water that you get from the store, get your water tested and if everything's all good, you can put him back into the 10 gallon. That way you'll know for sure that everything is on track. I know you said that your parents had the water tested some time ago, well you still need it done because water supplies can change. What may be bad a long time ago, doesn't mean it's all fine now.
As for names, how about Ziggy, Blaze, Titan, Neptune, Cosmo?