I'm just wondering if it's possible to use a cyclometer on the rear wheel of a bike that's attached to a bike trainer, and still have it record distance. The last one that I purchased (at Canadian Tire) didn't work that way, and I was just wondering whether it worked that way for all cyclometers.

All you need is a computer that takes its measurements from the rear wheel. Some computers are designed this way - otherwise, you may have to reposition your existing one so that the sensor reaches the rear wheel.

Could you provide suggestions of models that could do that (<$30). You also mentions repositioning, how would one do that? Sorry, I'm still pretty new at cycling.

If you can't afford a wireless computer - then you need one that has a wire long enough to have the sensor mounted on the chainstay (the horizontal tube the goes from your pedals to the rear wheel). Then it depends on where you want the computer itself.

For example, on my wife's mountain bike, the computer had to be mounted on the top tube (the tube that goes from your handlebars to your seat) in order for the sensor to reach the rear wheel.