Hexhamshire is the name given to the district around the town of Hexham in Northumberland. It was originally granted by Aethelhryth, queen of Northumbria to the Archbishop of York, but became part of the endowment held by the Bishop of Hexham when that bishopric was created in 678. This lasted until the Hexham diocese was united to that of Lindisfarne in about 821. It was then held by the Bishops of Lindisfarne and their successors the Bishops of Durham, until Henry I restored Hexhamshire to the Archbishop of York.
The 1911 Encyclopædia Britannica explains that;
The archbishops appear to have had almost royal power throughout the liberty, including the rights of trying all pleas of the crown in their court, of taking inquisitions and of taxation. In 1545 the archbishop exchanged Hexhamshire with the king for other property, and in 1572 all the separate privileges which had belonged to him were taken away, and the liberty was annexed to the county of Northumberland.
The term Hexhamshire is still in use today to refer to the area the parishes of Hexham, Allendale and St John Lee.
http://communities.northumberland.gov.uk/hexham.htm
http://66.1911encyclopedia.org/H/HE/HEXATEUCH.htm