The vital records of Southborough list him as HENERY. He was born on 31 August 1836. NEHGS lists this vital record in the Massachusetts Vital record to 1850, V1: 45 for Southborough:
HYDE Henery Jewell, s. Abraham and Elener, Aug. 31, 1836.
This is important for two reasons, the first being that I have not found this name written as Henery and I still have no idea where the Jewell came from! I have found a descendant online and I hope she will be able to tell me more about this. The second reason is that it is clear from this that he was born in 1836. As he is first born, it is important to look at the parent's marriage date. (Ahem...) Abraham and Elenor/Elener/Eleanor were married in Framingham on 7 May 1835. This is a plausible date for a first born child.
Henry's gravestone doesn't list any months or days on it. Here is a cropped shot of the dates. Notice that the birth year says 1837.
The next child born in this family does share a gravestone with his family but there are no dates on it for confirming evidence.
Henry married Amanda DAVIDSON on 1 May 1861 in Southborough. Now, that you know that, we can take a look at the whole gravestone.