What would John Murray drink?
Even though he wasn't given to liquor, I can't imagine he'd say no to a pint from a hometown brewery. Alton, in Hampshire, after all was known for its brewing and hops production.
Well, there's a modern (micro)brewery there. Real ale. Lovely stuff, I bet, with names like "Alton's Pride" and "Stairway to Heaven" (the second alluding to Led Zeppelin, rather than, say, the beatific vision).
The site's all in Flash animation -- not my favorite -- but it gives me something to look forward on that day I make it to that down some forty-five miles from JM's later home, London.
Of course, I don't believe there are any remaining breweries from his East End haunts of the 1760s, micro or otherwise.