I would try to fix it, sell it, or recycle it. In that order.
Try looking into fixing the iPod if your up for the task, since like you said there should be nothing really wrong with it other than a super dead battery. Probably the only thing you would need to buy to fix it is a new battery and a few simple tools to aid in opening the case. I recommend looking at the Repair Manual for your iPod from iFixit. Once you fix it, you tremendously increase its value and make it a better candidate to sell to people who don't like fixing things themselves. A 20GB iPod still holds its own in 2012 for playing music, and there are a lot of people who would find this device useful, and hence the parts as well even if you don't fix it. Its worth much more than most small electronics things going to be recycled.
After fixing it or not you could try to sell it on eBay, a yard sale, flea-market, Craigslist, etc. Working today March 16, 2012 its worth about $50 USD and for parts about $10 according to recent eBay completed listing prices. And again if you don't fix it you can still try to sell it for parts.
However if you don't fix it that may not be a good idea if you have personal information still stored on the iPod, eg address book, photos, etc. Because you will not get a chance to erase it. But if your not worried about losing any personal information from the iPod then by all means you can just sell it as is without having to break it open.
And lastly if you just don't want to deal with it anymore than just getting rid of it. Check into your local responsible recycling options, you may be able to simply drop it off at a nearby retail store or community waste disposal location that accepts electronics waste. Contact the people who take away your trash and see what their options and recommendations are. If your in the USA you may find the following local recycling links from Gazelle.com useful.