There's a huge Olympics superstore in the middle of Copacabana beach (in a big tent-like structure, IIRC roughly across the street from the Marriott) for official souvenirs, although some of them seem expensive for what they are. Souvenirs are also sold at the airport.

If you're in the Copacabana area, be aware that it's considered somewhat unsafe to walk along any of the side streets at night. Stay on the big street that runs along the beach and don't walk a couple blocks away from the water for vendors.

In Ipanema, the main shopping street runs parallel to the beachfront, two blocks back from the water. This would be considered a bit more safe, but you need to be cautious.

The museums I visited have gift shops, although the stuff at MAR (which has a store that appears to be accessible even if you don't pay to enter the museum) seemed expensive; much of it was like normal art museum gift shop stuff that you can buy almost anywhere.

People talk about a huge new modern upscale shopping mall in Barra, although it's not close to the Olympic venues. A second such in Barra focuses on home things.