Any inexpensive Windows laptop (since the cheapest Macbook would be around $800 I think) should work fine for something like Overdrive/Audiobooks.
Here are a couple of examples that should work great...
http://www.bestbuy.com/site/hp-15-6-...&skuId=4481400
http://www.bestbuy.com/site/asus-15-...&skuId=4831400
Or here is one that's a bit better (more memory, a better processor)...
http://www.bestbuy.com/site/dell-ins...&skuId=4815400
I got a very similar model to the third option for my mother as a $300 Black Friday deal and she loves it (uses it for email, web surfing, word processing, etc.), but it's a little more powerful than you'd absolutely need.
Depending on budget another option would be to find a used laptop that's maybe a year or two old as anything remotely modern should run OverDrive's software fine. Pawn shops and small local computer shops are sometimes a good place for decent deals if you decide to go the used route. Also Goodwill's auction site (or even a local location) can have deals on laptops...
http://www.shopgoodwill.com/listings...6&ending=Items