La Isla restaurant on Washington and 1st Street, delicious Cuban http://www.laislarestaurant.com/
2 Italian places I like, Margarita’s on 8th and Washington, BYOB. And Leo’s Grandevous, if you want that Sinatra nostalgia, cash only. http://www.leosgrandevous.com/
In Jersey City
Explore the restaurants and snack shops of "Little India" on Newark Ave, between Summit and Tonnele, with many delicious dhosas and snacks to be found there
Nha Trang Place, 249 Newark Ave for Vietnamese.
Ali Baba on Grove Street for falafal
Taqueria on Grove and Grand for, you guessed it, tacos
There’s a website dedicated to downtown bars and restaurants, you might find helpful http://www.kannekt.com/15/bars.htm
In the Jersey City Heights neighborhood La Gloria de Mexico is worth a visit (on Congress and Palisade) and on Central Avenue a Peruvian restaurant called Ceviche is good.