] Could you say what those two words are and your source? I'd like to have some more info. Thanks.

Maybe I can answer this, in part, anyway. The ending ley is early-middle Anglo-Saxon and means 'clearing in a wood', in which case you could suppose Bing to be a person's name, but not necessarily. If, as Chloe claims, it isn't ley but ey meaning 'island' in Saxon, then you have a totally different kettle of fish.

The only book about place-name theory that I have is What's in a name? by C Stella Davies and John Levitt, which is great for explaining how place-names worked, but doesn't include every single example in the British Isles and doesn't include Bingley, so I don't know which ending is more likely for the town of Bingley in Yorkshire.

I'd also like to point out that study of Anglo-saxon names in JA's time wasn't at all well advanced, and I think it's highly unlikely that JA chose the name depending upon its meaning in Anglo-Saxon. It's far more likely that she used the name of the place, and as Anielka has said, got it from her various ancestors, or from her acquaintances.