'Edgware' means Ecgi's weir, the fishing pool of an Anglo-Saxon chieftain.
It was first recorded in the 970s as Aegces wer; the weir was a pond where Ecgi's people caught fish, and it was likely built on one of the brooks along the parish boundary. A legal record of 1422 has Eggeswer, and by 1489 writers had added the 'd', making it Edggeware.