Yes, there are several responses to this presumably double request on the Cravings board, and though they were worded only slightly differently, I did get different impressions as to what the OP was looking for. I suggest those responses either be moved to to this thread, or both threads be maintained as separate topics.
I also recommend Campanile because the food is excellent and the prices moderate, but I found another Italian place, though it is a bit outside the area you asked for, that I liked very much. It is called Nino's of Tuscany and is on 58th street around 6th or 7th avenue. There is a pianist called Irving Fields, he is 93 years old, and he plays there 6 nights a week. The food is not too expensive and it is a very nice restaurant. The service is excellent. It is not wanna be actors and actresses that work there:)
Presuming that by "authentic" your relatives mean traditional Italian...
Campanile is a neighborhood gem, hidden away on 29th St., b/t Madison and Park Av. S. The food's well-prepared and delicious. Service is welcoming and capable. The space has a pleasant, old-fashioned feel with comfortable seating and a conversation-friendly noise level. Prices are moderate.
In the Times Square area, Trattoria Trecolori is a favorite of mine and many other Hounds. Very tasty homestyle fare is served by a friendly, efficient staff in a space that has well-spaced tables and attractive contemporary ambiance.