I believe it would,as it would stir true rivalry. I live in 15 minutes from stadium. I find calling a NJ team a NY team phony,disingenuous and not helpful in creating a true tribal experience for fans in this MLS 3.0 . Not many if any fans from NYC go to games,they are all NJ residents. The stadium is located in a perfect location,surrounded by Portuguese,Brazilians,Equadorians,Columbians who are soccer mad, yet few go to games.
I don't think changing the name to NJ Red Bulls or just Red Bulls is going to cause city folks (manhattan, Staten Island, Brooklyn, etc..) or NJ folks to decide "Oh ok. I'm going to go watch them now"... There are 2 ways in to increase attendance: 1) Big name signings and 2) attractive soccer. Right now the RBs are lacking both. (Throwing everybody forward into the attack with no idea how to break down the defense only to expose themselves to counters is not attractive soccer).
The City fans are the abused children of MLS, and we should pity them, but I don't think a name change will make much difference in attracting new fans. More aggressive marketing would help, as will taking a more serious approach to signing new players.
I also think giving more benefits to season ticket holders should increase attendance (if it in fact increases season ticket sales). Example: Offering more rewards and giving more Members+ reward points. What does it cost to have the players sign a few extra jersies/memorabilia to auction off? Offer more gift shop items to be redeemed with Members+ points. Allow season ticket holders to trade Members+ points for concession vouchers or add money on their card to be redeemed at concession stands.