The two closest areas are Kalithea and Afandou. Kalithea is to the North side of Faliraki (towards Rhodes Town) and Afandou to the South. I would say Afandou is more likely on a cheap deal, Kalithea tends to have the smarter hotels, Afandou lots of small self catering apartment blocks. You never know your luck though!
Kalithea is mainly based around the hotels, there's not a great amount of bars, restaurants, etc in the vicinity - handy for Faliraki though. Afandou is a traditional greek village but does cater to tourists with it's shops, bars and restraurants. Both are fairly quiet and laid back resorts but only a 10 minute taxi ride from Faliraki, if that.