My dad is in his 50's. After he runs his hands swell up. What could be the cause of this? He is slightly overweight. Runs 3 miles every few days.

Swollen hands. Swollen hands after running could be due to increased blood and fluid flow to hands as a result of an increased heart rate, which helps the muscles contract and move efficiently. Other causes include dehydration, running outside during high temperatures, and fluid retention as a result of a high-salt diet.
Not completely known. Hand swelling during exercise is common. Exercise increases blood flow to your heart and lungs but reduces blood flow to your hands, making them cooler. The blood vessels in your hands sometime overcompensate by opening wider, which sometimes leads to hand swelling.