An update, in case anyone else is interested in this line. I have located probate records for the estates of both Thomas William Turner and Henry Blunt Turner. Thomas William died only a few years or so after Henry B. Turner. Thomas William was originally appointed guardian of Henry B's young son Simon, and after his death that role was passed to Robert Cotton Turner. Francis A. "Frank" Turner, also a son of Henry B. Turner, acted as an executor of Thomas William's estate. Much of the administration of Henry B's estate was handled by Thomas William. So, it's certain there was familial relationship here, and gauging by the age of Thomas William and his sharing of a home with Frank Turner in 1850, it seems most likely he was indeed a son of Henry B. Turner.
William's estate records show payment to his ward Simon Turner, as well as settlement for his own two children, Joseph Alden Turner and Mary W. Turner (who later married John Bruce Carson). In Henry's estate record, it shows Robert Cotton Turner being nominated to be guardian of Simon in 1860, the year after the death of William.