Is it anything to do with the card game Euchre ? This game is played a lot in Devon and also, I believe, in some areas of America. I don't know how to play but have watched them play down the local in my wifes home village. One of the cards is called a 'Benny' - I think that it is some sort of wild card.
Click http://www.m-w.com/cgi-bin/dictionary?book=Diction ary&va=benny&x=18&y=17
and the link will take you to the Merriam-Webster (ie American) Dictionary's website's page about the word 'benny'. It seems pretty clear that Thewillow's idea is correct. The answer, presumably then, is "swallow it"!
The Merriam-Webster Dictionary, an American publication. gives benny as meaning a tablet of amphetamine, . a stimulant. It must derive from 'benzedrine' That is certainly what a benny is in the UK..
Interesting to hear FredPuli's suggestion that a Benny is a pill in England - I've always used it to mean "a prat". For example, "James is a bit of a Benny".
Benny, the famous character in Crossroads, the long running TV soap, was a simpleton. So calling someone 'a Benny' in the UK would mean that they were a bit simple-minded or backward mentally or 'a bit of a prat'.