I've been living and working in Tribeca for over 25 years, which qualifies me... well, who knows if I'm right, but I think by far the best quality:price place in Tribeca is Gigino, on Greenwich St (they also have a second, fantastic location in Battery Park, with great harbor / Statue of Liberty views,).
Gigino offers fresh, well prepared food, a large menu with lots of daily specials, pretty good brick oven pizza, and a decent enough wine list (you can also byob for a $15 corkage). A house special called spaghetti al Padrino is our perennial favorite - and a dish we've never seen anywhere else - spaghetti with beets, escarole, anchovy, parsley - really very very good. It's a fairly large place by NY standards and lively without getting deafening; the service is almost always very good.
BTW: we avoid Roc, neighbor to Gigino; unless you care to over-pay for lesser quality served with plenty of attitude.