My grandfather worked at Somerset House during WW2. He was with the Ministry of Supply, which took the Inland Revenue's place there, after they briefly moved out of Somerset House.
I believe there were apartments at Somerset House, which were available to the on-site Office Keepers. For example: the apartment available to the Office Keeper for the GRO consisted of 2 sitting rooms and 3 bedrooms. Other apartments offered a similar amount of rooms (if not more) plus a kitchen.
Haven't worked out yet what happened to Richard L T Hill's parents. I think his mother's maiden name was Proudman. Richard is living with the Proudman family in the 1841 census in Tamworth. And he might be the teenage Richard Hill (clerk) who is lodging with a family in Birmingham in the 1851 census.
UPDATE: Richard Hill snr died 1836/1837. His 1837 will (with images) is on FMP. He mentions
his father, Thomas Hill (gentleman), of Drayton Bassett, Staffs.
He also mentions his wife Mary Anne, and son Richard L T Hill. Also mentioned is Richard snr's brother, John Hill (butcher) of Tamworth. Not sure when Mary Anne Hill died.
Now here's Richard L T Hill's connection to the Janet's father, Thomas Hill, who works at Somerset House:
Janet's father is the
second son of Thomas Hill, gentleman. This is confirmed in Thomas Hill snr's 1849 will. So that makes Richard L T Hill and Janet Hill cousins.
Also noted in the 1849 will is a Richard Lateward Taylor (brother-in-law of Thomas Hill snr, gentleman). So he would be Richard L T Hill's great-uncle and namesake.
Based on Thomas Hill snr's 1849 will, it sounds like the Hill's were a rather prominent family in Staffs. It'll be interesting if your Dix chap is a Hill descendant.