Ask the elders. Always a wealth of information, question the seniors about the sorts of games they used to play when they were children. Get suggestions from them about the sorts of games they think you should design.
Design games that require memory skills. Get 25 to 30 stones -- buttons or coins also will do -- and devise a pattern. Have the players study the pattern for a specific period of time, and then cover it with a towel. Each player has to reproduce the pattern as closely as possible. If you don't have enough pieces for every player, take a digital photograph of each entry and compare them later.
Devise games that teach traditional skills. Native Americans had to be skilled at being stealthy. Appoint a "keeper of the fire" and blindfold her. The idea is to creep up and "steal" a piece of wood from the pile in front of her. The firekeeper has to keep her hands in her lap unless she reaches out to touch a wood "thief."
Design your own board game. Think of a Monopoly board or snakes-and-ladders as a basic idea, but then design it to fit a Native American context. Make cards with things like "go back five spaces because you didn't snare a rabbit" or "advance to the happy hunting ground" and play the game with dice.
Include team games that require physical stamina. Position the students in different places along a road where a runner has to pass an object --- a peace pipe or a plastic tomahawk to the next person in line. If you don't want to go on a road, use the track at school, and line up the members of the team and tell each one to do one or two rounds of the track before passing the object to the next person.