Name of third-born registered in Somerset House (4)
I believe the answer is:
seth
'name of third-born' is the definition.
(Adam and Eve's third son)
'registered in somerset house' is the wordplay.
'registered in' says the answer is hidden in the clue ('registered' as in 'seen').
'SETH' can be found hidden inside 'somerset house'.
(Other definitions for seth that I've seen before include "Adam and Eve's third son" , "Third son of Adam and Eve" , "One of the first names recorded" , "Younger brother to Cain and Abel" , "Indian writer" .)