Battery Health
A measurement of the phone's current battery capacity. After many charge cycles, the maximum amount of energy a battery can hold diminishes. Example: a phone with 85% battery health means that when the phone is charged to 100%, it's charged to it's full capacity, which is only 85% of the original capacity, and thus real world battery duration will feel shorter.
Recommendation: Buy your phone with no less than 85% battery health.
Reason: With less than 80% battery health, the performance of the phone can start to slow.
Another Shopping Strategy: Most phones can have their battery replaced for a fee. Find a great deal on a phone with 75% battery health, purchase that phone and get the battery replaced. Check battery replacement costs before purchase.
Parts
Phones break, and people fix them for longevity of the device. When buying a secondhand phone, be sure to find out if any parts have been replaced. Ask where the parts were replaced and do your research on the company. Sometimes if a part is replaced by someone other than the original phone manufacturing company, the phone may report that "one of the parts in this phone is not original"; there's nothing wrong with this per se, but some people may not like it which means it will affect the resale value down the road.
Condition
Do you really need a phone in mint condition? Probably not if you use a case. Secondhand phones in used condition with a couple scrapes or scratches around the body are typically harmless to the phone's function. Scratches on the screen are more important, as a deep or large scratch can affect viewability, even after applying a screen protector. Take your time and make a good judgement call based on the information you have on the phone. Ask for more photos from the seller if you're even a little bit unsure. Don't jump on a deal if you're unsure of the phone's condition.