'The House' is the name sometimes given to Christ Church College, Oxford; it derives from the ancient description of the college as 'The House of Christ'.
'Mercury' is the fountain in 'Tom Quad', the quadrangle named after the bell 'Great Tom', which is housed in the tower above the entrance to the college.
The statue of Mercury presently in the fountain is a copy of one by the sculptor Giambologna (1529-1608), and is thought to have been placed there in 1928 (after the events of 'Book 1' of Brideshead Revisited). The head of the original statue, which dates from the time the fountain was built in 1695, is now kept in the Upper Library at Christ Church.
Traditionally, college 'hearties' (sporty types) would throw 'aesthetes' (artistic, sensitive types) into the fountain; hence, the reference to Anthony Blanche being 'in Mercury'.