I think a year is a reasonable timeframe, if the child wants it enough - but the child really will determine this.
One of most the challenging things to teach is persistence - instead of flitting from one thing to another each time something is boring or difficult, doing something long enough to become good at it. A common approach is not dissimilar to this: to sign your child up for something for a long period of time (like a year) and then tell them they have to do it the whole year.
This rarely works, if it is an edict from above, because the child is then just doing what they're told to do - not doing what they choose to do. They learn that they have to listen to mom and dad, but nothing else of value.
What works, though, is when the child decides to make the commitment. If the decision is theirs, and it's something they're excited about, they're much more likely to persist - and to learn how to do so.
Music is a very common choice here, because it's something many children really do want to do, and want to get good at! But, it's simply impossible to get good at anything musically in three months - it takes a year before you're really doing anything interesting, for the most part, with nearly every instrument, if it's being taught well.
So the approach here is to sit down with him and lay everything on the table. Tell him he can do this, you can pay for the lessons, but only if he promises to stick with it for the year. Explain why it's so long - because it will be somewhat boring for a while as he learns basic technique, and it may well be close to a year before he's playing the really interesting songs.
Treat this like a contract negotiation - don't just get an "I promise", but set out what the rules are and what the consequences are for breaking the rules. Are you okay with bugging him to practice constantly? Or does he need to manage that himself? What's the minimum practice schedule you can tolerate? (Typical 8 year old practice schedule would be maybe 20-30 minutes a day, 7 days a week - but that's very hard to keep up all the time). What happens if he does give up after three or four months? Any contract has to allow for the possibility of breaking - what can you and he agree on here?
If you treat him like an adult here, and let him make the affirmative choice to do this, and then support him - there's a good chance it will go a long way to teaching persistence, and many other things that come with learning music.