Sure!

1. Is your staircase straight, or does it curve around? It can be pretty difficult to find a secondhand curved lift that fits your staircase - a straight is much easier.
   
2. Related to point 1, but try to find a lift that fits to the same size staircase as the one you own. Not a lot of stairlift companies (in my experience) will sell you a bit of rail. Rather, they'd charge you for the whole machine.
3. If your staircase is outside, remember that you'll need an outdoor lift, and not an indoor one. Outdoor lifts are designed to be weatherproof, and this'll save you a lot of heartache later down the track (pun unintended) when you have to deal with rust.

I also recommend that, when you start shopping for se one hand lifts, you grab the serial number from the machine and call the company that supplied it. Sometimes, they can give you some good advice on that lift (like how old the machine is, if it's an indoor or outdoor model, etc.).