Right-click on the mine tile before you build and it tells you your expected income. Some come in around 10% return each turn, so the payback is in 10 turns. I always build those. Some are closer to 20 turns, and I build those when I have a surplus.
Plus, you can put a merchant on a developed mine and often get very good income from them, which can cut the payback time by almost half.
I have a 2,000 florin silver mine in Northern Italy ... it's generating 200 florins/turn and an additional 159/turn from a third-level merchant, so 2,550 (including the merchant) divided by 359/turn income is six turns before I'm running clear profit every turn.
One more point ... it's worth micromanaging if need be to keep architects in big cities once they show up in your retinue ... a 10% discount on building costs adds up over 20-30 turns if you're using them to build 4,000 florin improvements instead of 600 florin roads or crop rotation.