Edgware in Middlesex was previously called Edgworth. Various historical documents refer to it as such, and it was known as Edgworth in the time of Samuel Pepys, who mentions it in his diary. Also, it is marked as Edgworth on old maps.
This, of course, does not by any means confirm what Richard Edgeworth recalls hearing as a boy from his grandmother, Lady Edgeworth, as recounted in the “Black Book”; which is where the idea of coming from Edgworth in Middlesex comes from. But it does mean that Edgworth/Edgware is as likely to be where we take our name from as places of similar names in Lancashire, Gloucestershire and Bedfordshire. Interestingly, Richard Edgeworth mentions alternatives himself in the “Black Book”.
However, I think it might be a mistake to think that the Edgeworths resided at a place of that or similar name immediately before Edward and Francis went to Ireland. We have some information about the two or three generations before Francis, but it is sparse and disjointed.
I hope the above is of some interest to you.