The minimum requirements (as well as any optional features) depend on what you're working on, and how deep into it you're going.
For example, if you're working on vector games, you want an analog scope with 2 channels and XY mode, for displaying vector graphics (which don't look as good on a digital scope). But you won't use XY with JAMMA games.
One channel is enough for a majority of general debugging (as much of the time you just want to see if a signal is present, and if it looks ok). But there may be times when you want to see two channels, when you want to compare timings of two different signals. Very rarely do I personally need more than two for arcade use, but that's not to say it can't happen. I have four channels on both of my scopes, but never really use more than two.
As for bandwidth, I can't speak for what speeds the newest games run at, but for 80's stuff you'd want at least 50 MHz, and if you can find it affordably, go for 100 MHz or more. Again, it comes down to having the extra power for those times when you might need it. The more bandwidth you have, the more clearly you'll be able to see signals, but how much you need, and how clearly you need to be able to see any given signal is going to vary, depending on the situation. So it's better to have too much than too little.
If you want a decent, affordable, modern digital scope, a lot of folks have the Rigols. I've been considering picking one up myself, for the digital features. I'm not up on which is the latest model, but there's one from a while back that was hackable, to expand the bandwidth. Digital scopes have bells and whistles like automated measurements and cursors that can be useful sometimes. See here:
https://www.youtube.com/watch?v=LnhXfVYWYXE
But in the end, they're all tools. And when you're troubleshooting, you never know when a better tool or more capability might come in handy, so it's a matter of having as much in our toolbox as you can reasonably afford (and knowing how to use it, which there are many good tutorials for on youtube).