This question has been asked a couple of times before, and two names have been supplied by u/Georgy_K_Zhukov.

They are those of Peter Conover Hains, an artillery officer who fought at Bull Run and Vicksberg, and survived to serve as head of the Norfolk engineering district in World War I, and William Harding Carter, who, although not formally enlisted, was a child dispatch rider during the civil war, subsequently entered the army, and enjoyed a 50+ year career that resulted in him reaching the rank of Major-General, and ended with his involvement in training troops for World War I. He eventually retireed in 1918.

You can check out the original threads on these two here and here.