Should I jailbreak my phone or unlock it?
I'd do neither, personally. I don't trust or like any unlocking service or jailbreaking, so I would try to check if it's carrier unlocked by this (optionally not contacting T-Mobile to ask them yet to unlock). 


https://support.apple.com/en-us/HT201328


If it's locked, I would take the phone to a T-Mobile shop and tell them truthfully  the essential facts you know:


An acquaintance said they couldn't afford this phone and I'd like to see if I could establish service with it. Can you help me shop for a mobile plan?


You don't have to tell them who / relationship and certainly don't speculate. If they are probing to see what else you know about the phone, don't get nervous and do be polite and deliberate. Just repeat what you told them in pretty much the exact same words and then ask if they can help you. Just keep in your mind - you're notthe previous owner and you're the owner of the device now and this is a chance for them to earn your business. 

It's not their "business" if you paid for it or not and they shouldn't tell you anything about the previous owner. It is their business to help you pick out a plan and understand if it has a contract or not and propose another plan if you don't like the one they suggested initially. From some very astute comments, you should probably protect yourself with a written bill of sale if you intend to say you "own" the device. Paying a dollar and writing up a transfer keeps you from lying if you go to the store. If it doesn't work out, you can always sell it back later to the previous owner.

If you can get service in your name and avoid a contract, I'd do it. Then you can look at getting it unlocked after a month or two if you're not happy with your service in getting the phone working and with the service as a user of T-Mobile. At that point, they'll have a much harder time saying you're to be penalized as a paid up customer asking for an unlock.

Make sure that whomever signs into this iPhone doesn't keep "Find My iPhone" turned on. If the device ever gets locked and the user doesn't know the password or can't get the verification code to the account. You will be stuck with an expensive coaster that Apple won't be able to help you with. They will need the proof of purchase and that doesn't exist since only T-Mobile has that and will not give that up until the balance is due.