Well, a start point would be:
http://www.aluguetemporada.com.br/ferias/brasil/rio-de-janeiro-estado/r29369
Beyond that you can always try:
http://www.zapimoveis.com.br/
Some pointers...
Most rental agreements are for significantly longer than six months, unless you are looking at short term, holiday rentals, inc which case there is usually a premium to be paid as it's a renter's market and there isn't much incentive to provide anything cheaper to short term tenants. Rio is not a cheap city to start with btw. Of course if you are fluent in Portuguese and possess a little charm you might be able to swing a better deal on the grounds of an assured rental period longer than the typical tourist.
Have you been to Brazil before? Do you speak Portuguese with any particular fluency? Are you familiar with the city? Your comment about renting in a favela suggests otherwise.
To be honest, rental prices are much of a muchness between Copacabana, Botafogo, Leme and most of Zona Sul. Ipanema and Leblon being usually a lot more expensive. Tijuca isn't especially cheap either on the whole. You can still find reasonable options if you dig hard. Familiarize yourself with the terms kitnet and sala conjugada if you are on a budget but beware that may mean the physical inability to swing a cat.
Outside of the traditional tourist zones, you will find it harder getting a short term lease but rental prices are cheaper. Good locations, with good or at least reasonable amenity that I would recommend would be Vila Isabel, Méier or Rio Comprido. How much further you venture, would be conditioned by your comfort factor and whatever reasoning you have to stay over that length of time. Your best bets as with most things are personal contacts. If you have friends or associates in Rio ask them if they know of anyone looking to rent.
I'd ditch the idea of staying in a favela if I was you, unless you know someone who lives in a particular community that's prepared to share and even then not all of these communities are the same in terms of amenity or security. You 'll struggle to find rental listings anyway.
Good luck! You'll need it, especially if you don't have a large budget.
