1. Avoid peak tourist times and areas. Accommodation will be your biggest expense. Avoid high season (Christmas, New Year and Carnival) and don't stay in the tourist areas. Standard of hostels can also vary greatly for the same price. A good hostel I found is Cabana Copa - clean, tidy, right near Copacabana Beach and even includes a great breakfast.
2. Eat like the locals. There are plenty of cheap restaurants where you can pay between 5-10 Reais (and sometimes less) for a huge meal. If your appetite is not big you can probably share a meal with a friend and halve that price. Most of these restaurants are easily identified as they have blue or yellow plastic chairs outside and are filled with locals.
3. Don't pay hefty club entrance fees. Avoid paying 20-30 Reais to get into a club. Instead go to one of the many roadside bar/restaurants as mentioned above. Many have nights where they have live musicians play and you can join the locals dancing samba.
4. Take the bus and subway. I read many things telling me how dangerous it was to take these forms of transport at night and advising me to take taxis. Unfortunately taxis in Rio eat up your budget very quickly as many drivers seem to like to take the long route to extract as much fare from you as they possibly can! I had no problems using buses and the subway. Of course as always when travelling common sense prevails. Only travel on buses and subway dependant on your route and destination and as long as people are around and you feel safe. Around 2 Reais for bus or subway ride is better than an expensive taxi ride.
5. Climb Pão de Açúcar. You can actually walk up to the first stop on this famous tourist attraction. It costs nothing and you get a fantastic view. Take the bus to Praia (beach) Vermelha and you will see the park at the foot of the hill.
6. Walk up to see Corcovado. This is definitely more of a climb but you will pay just 5 real to get in to see the Christ statue as opposed to the 36 Reais ride to the top plus entrance fee.
7. Eat up large. Make the most of the hostels that offer free breakfast!
8. Take a friend to the beach. Share the 5 Reais cost of the umbrella plus have someone to look after your stuff while you are in the water.
9. Take your own snacks to the beach. Otherwise you will pay twice the price.
10. Try something new. Don't be afraid to try local food - it's delicious. For example avoid buying expensive potato chips and instead eat the local Globo snacks made from cassava flour. Tasty and unique to Brazil!