After a sprain, which ankle rehab exercises (done at home) will help prevent another sprain?

Proprioception. Id google for proprioception exercises. They are balance type exercises and will help more than anything for this type of injury. Strengthening and stretching will help as well. However, once the ligaments are sprained, they tend to heal in a stretched out position, and you are at risk for recurrent sprains.
Tough to prevent. Ankle sprains typically happen from a forceful injury and are difficult to prevent. Rehab exercises at home to recover range of motion can involve toe raises, "writing" the letters of the alphabet with your big toe by moving and rotating the ankle. If there is a chronic instability or the feeling of "weak ankles", then there may be a mechanical abnormality and you should have it checked out.
Yes. Rehab exercises are designed to strengthen the muscles and ligaments that provide stability to your ankle. The stronger they are, the less severe any sucessive injury to the same ankle.
Do the alphabet. Move the foot and ankle and draw the alphabet with your toes. Start lifting mild weights with your feet to help strengthen the ankle.