Do you have Jacob Eaton's birth certificate?
March quarter 1842, Bedford reg dist
In 1851 he is 9 yrs old, son in the household of Jacob Eaton, 56.
I might wonder whether he was actually the grandson of that Jacob, and possibly the child of an unmarried daughter. His father's surname might have been Smith, and he might have gone by that name but used Eaton as his "official" name for registering events and in censuses. We've certainly seen other instances of that.
In 1841 (the family is "Caton" at Anc'y and has been fixed by a user), Jacob and Mary Eaton do have a daughter Fanny (mistranscribed Hanny) who is aged 15+, and they likely had older children as well. Mary likely died in 1844 (two Mary Eaton deaths in Bedford that year), and was probably 40+ in 1841 (transcribed as 46 at Anc'y), so Jacob the younger could certainly be her child, though. But a Fanny Eaton died in Bedford in 1846, so I might still wonder whether he was reared by his grandfather.
How about his marriage certificate -- did he name a father? Not that that would be conclusive; maternal grandfather was a common choice for a child of unmarried parents in that situation, especially if the child was reared in the mother's family.
In 1901, he and his wife and children are all "S Eaton". That does suggest that they used the surname Smith. And being born to unmarried parents is about the most common reason for name-switching like that.