This is interesting - not a single Icelandic source has any information of what became of this particular Jón Gíslason - he just disappears from the records.
What is the source for the Doctor Jón Gíslason in Norway being exactly this Jón Gíslason and not someonne else?
Anyhow, regarding his parents, nothing whatsoever is in the database about their ancestry.Krístín Bergsteinsdóttir was probably the sister of Sigríður Bergsteinsdóttir who died at Mógilsá around 1750, and their father *might* have been Bergsteinn Gíslason, who is in this area in the 1703 census, but that is only speculation, nothing more.
As for Jón's siblings, his parents had twins, Bergsteinn and Guðrún in 1747, who died shortly after birth, a sister, Guðrún, born 1748, who also disappears from records, a son, Jón, who died a few days old on Nov. 10th, 1749, a child born in 1754 who died before baptism and a son, Jón, born in 1761 who died in 1775.
In other words, there are no known Icelandic relatives in the Islendingabok database.