Henry is probably a corruption of Honoré, which was the dit name associated with the family. It's hard to say whether it was Anglicized by the family, or written down incorrectly by the census taker. The census record doesn't link back to an image for further inspection.
Edit: Just re-read your description pointing out that descendants are named Henry. So it seems that this was done to Anglicize the name.
The dit name Honoré comes from the founder of the Tison/Tesson/Theson etc. lineage. Again, we have problems with duplicates, but you can find him under Honoré-Louis Tesson-81, Honoré-Louis Tesson-83, or Honoré Louis Tesson-61.
Given all the duplicates, sourcing issues, and some date discrepancies, I'll try to spend some time this weekend sorting things out a bit for this family.