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 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.