Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
AGATE
As described in historical dictionaries
Patronymica Britannica (1860)
AGATE. At-the-Gate, of some town or forest ; less probably, a sobriqixet ' applied to a diminutive person, in allusion to the sniall figures cut in agate for rings.' Nares and Halliw. in voc.