OK, we did some research.We are fairly certain (but not absolutely sure) (based on church records, the 1835, 1845 and 1860 censuses) that this Jón is the one born in 1823 in Setberg-parish, the son of Jón Guðmundsson (b. 1782 and Guðrún Sigurðardóttir.
Jón Guðmundsson was the son of Guðmundur Jónsson and Valgerður Guðmundsdóttir. They were at Arnarstapi, Laugarbrekka-parish at the time of the 1801 census.
As for Guðrún Sigurðardóttir, we have not yet been able to identify the right one.