Hi Oren K!
First of all i think you should check that is neon is your problem or not. To do that you should turn on you circuit but do not turn on neon by that circuit. then you should bring another neon close to you circuit and turn it ON. If neon is emitting any EMI it will reset you circuit otherwise you should concentrate on improving your circuit's supply. i suggest to use seperate 7805 for you MCU with 10 - 100uf at its output.