Perhaps Henry's middle name is Adams. His mother may have died in childbirth or shortly after Henry's birth -- but this is just speculation because I do not know Henry's exact date of birth.
Henry A. Remington who married Sally W. Reynolds was identified as "son of William." I am making an assumption that he is Henry A. Remington, son of this particular William Remington. Their birth dates seem about right for this marriage (see their ages at death, in the RI Death Index).
Henry A. is mentioned as a son in the 1852 will of William Remington.
His death record at Providence also identifies him as "son of William."