Origin of the name:
Eyre:
1) The right of the king (or justices acting in his name) to visit and inspect the holdings of any vassal.
This is done periodically, usually at irregular intervals of a few years.
(MEDIEV-L. Medieval Terms)

2) English circuit court.
(Gies, Joseph and Francis. Life in a Medieval Castle, 230)

3) From the Latin iter, a journey; usually applied to circuit of royal judges.
(Frame, Robin. Colonial Ireland, 1169-1369, 144)

4) A periodic visitation of a county or group of counties by the king's court.
(Waugh, Scott. England in the Reign of Edward III, 238)

5) A travelling court of royal justices, periodically sent round the country in circuits to enquire into royal
administration and to hear both civil and criminal cases.
(Reynolds, Susan. An Introduction to the History of English Medieval Towns, 198)

Note: in itinere (Latin) = on a journey