Exchanging Li-ion batteries: what to take into account?
The main things of concern are, at least for lithium-ion single-cell batteries:

Ensure the new battery has the same number of cells as the old one--do not replace a single-cell battery with a 2-cell battery or vice versa
Ensure the new battery is the same chemistry as the old one (don't replace a LiFePO₄ with a standard lithium cell, for instance)
Ensure the new battery has the same or higher charge and discharge current ratings (that's current rating, not C number; you'll have to convert if they have different capacities)

Note that what really matters here is what current the battery actually is charged and discharged at, so if you can identify that, that is your real limit, not necessarily the old battery's ratings. It's unlikely the battery is being used right at its limit (and the new one really shouldn't be either!).



These are the only real safety concerns, but you probably also want to ensure the capacity of the new battery is about the same or larger. Otherwise you might find you have nowhere near the battery life you used to!