I happened to me the same with a wireless Logitech mouse. Just choose USB overdrive or single button mouse (even if it is a 3-button wheel one). Try with other brand/generic ones. Curiously a Microsoft wireless one work perfect, but the Logitech didn't.
I have noticed there are some Logitech OS9 drivers in the net.
Can't help the OPs problem as I don't use Overdrive, but I've had good luck with Logitech Mouseware when used in conjunction with a Logitech mouse. We frequently have big boxes of surplus OEM peripherals appear at work, and I've been know to go digging for Logitech OEM mice for that reason. I've found Dell and Gateway branded Logitech mice and all have worked perfectly in OS 9 with full 3-button capability using Mouseware.
Thanks for this post. 5 mins ago I didn't have a clue what overdrive was (with the exception of a gearbox add-on, on one of my first 70's sports-car)! Now with a brief visit to Macintosh Garden, all has been revealed, and it's a program I can see myself using in the future.