Compensation is tricky. You don't want to ask too much, and sounds like you're greedy. You don't want to offer too little, and sound like they can take advantage of you any time.
Furthermore you have to judge intentions. If your neighbor has offered to compensate you, it may be that they want to keep your relationship relatively impersonal. Or they might just be socially awkward.
One workaround for this, that can be used in almost any situation, is to suggest that you're giving them a "friends and family discount". This is especially appropriate if you are a professional who ordinarily gets paid for this kind of thing.
Hey John. I'm happy to help you out with your computer problems. Ordinarily I get paid about $130 for this stuff, but I'm going to give you my special discount. How about we go down to the local [pub | bar | craft brewery | etc] afterward and you buy me a couple of rounds, and we'll call it even?
Or pick whatever activity seems good for you, like asking them to pick up something to drink and you both can hang out and watch the game.
If you aren't a professional, then use that to suggest you wouldn't feel comfortable accepting money.
Hey John, thanks for the offer to pay me but since I don't do this kind of thing for a living I wouldn't feel right accepting money. How about instead you order us a large pizza and (something to drink) and after we can hang out and watch the game?
On a separate note: Like it or not, drinking alcohol is a generally approved activity in American culture. If someone has strong feelings about this, it's up to them to voice their disapproval in a socially appropriate way. If it's against their religious beliefs, they've probably had to do this many times and already have a script.
In return, you should never feel embarrassed offering alcohol to someone who doesn't drink. Just play off of it.
Hey, no beer, that's fine. Let's make it a root beer then. I know this place that makes a great craft root beer, you should really try it.
Either way, once you've made an offer you should just relax and wait for them to either accept, or make a counter-offer. It's up to them to figure out how much they are willing to barter vs. what might offend you, and you can relax knowing you'll probably accept whatever they say.