Hmm...I think the local angle should be played up, to differentiate your business from the pre-packed sandwiches anyone can buy almost anywhere. You'll need to give potential customers a reason to not simply buy a pre-packed sandwich from any of the numerous places nearby that will be selling them (unless your business is in a village and maybe not even then). Supporting the local economy could be that reason, for some people.
Maybe "<town name> Sandwiches" as the business name and something along the lines of "Freshly made from local ingredients" under the name on signs and adverts. Assuming they really are, all of them, so you're not courting trouble with inaccurate descriptions.
Fun names are all very well, but it might be more useful to get some clear reference to the stuff you're selling right there in the name. The Bait Box...a fishing supplies shop. Megabites...food of some kind, no idea what kind, probably a cheap and nasty takeaway that won't last long. Tacky name.
If you fancy a name that you'll need to explain to people: Tasty Trenchers. May as well get a bit historical - trenchers are the English precursor to modern sandwiches, essentially an open sandwich (the trencher was a large piece of bread used as a plate). Very common in the middle ages.