To begin, stand on one leg at a time and try to maintain your balance for as long as you comfortably can, first with your eyes open, and then with your eyes closed.
Next, straighten your up-side leg straight out in front of you for as many repetitions as you can comfortably manage – you can hold onto something with your hands to help you stay balanced as you do this. Strive to keep your down-side leg straight – this will help develop strength in the hip abductors of your down-side leg, called your gluteus medius and minimus – these are the muscles that are primarily responsible for keeping you balanced as you stand, walk, and run. You will feel these muscles get a good work out along the upper and outer aspect of the buttock of your down-side leg.
Next, balance yourself on one leg with your up-side leg extended straight behind you as high as you can comfortably take it. You can support your hands on something in front of you like a chair, table, or gym equipment. Repeat on the other side.
Then, try the same balancing exercise without support for your arms. You can keep your arms pointed down toward the ground or straight out above your head.
For an overview of exercise progressions that you can work at to improve your mobility and balance in a systematic way, please feel free to visit our Mobility Exercise Progressions page here:
Mobility Exercise Progressions
For some suggestions on how to set up a simple workout area at home, please feel free to view:
Suggested Resources for a Home Gym