This officer of the 15th Iowa was greeted by General Grant with a handshake and memory of the officer's father from the father's service in Mexico with General Grant.
William Worth Belknap. (Later briefly Secretary of War under Grant, before resigning in order to avoid impeachment for corruption. He was impeached anyways.)
William W. Belknap was an officer in the 15th Iowa and his father did serve with distinction in the Mexican-American War. Grant later named him Secretary of War, so he must have had a high opinion of him.
Can't find a reference to the handshake or any specific memory Grant may have had of his father, but I'll stick with William W. Belknap unless I find a better answer.
William Worth Belknap(1829-1890).
Source: Timothy R. Mahoney, From Hometown to Battlefield in the Civil War Era, (New York: Cambridge University Press, 2016), p. 249.
William W Belknap would later serve as Secretary of War (1869-1876) under President Grant at the urging of General of the Army William T. Sherman. Belknap served under Sherman in the Atlanta Campaign and fought with Sherman in his March to the Sea.
While doing a number of great things as Secretary of War, he was forced to resign for receiving profits in a traders scandal.