What does 'let the idea go begging' mean? Does it mean someone hesitates when making a decision?
The Free Dictionary gives the definition:
Fig. to be left over, unwanted, or unused. (As if a thing were begging for an owner or a user.)
They are citing the McGraw-Hill Dictionary of American Idioms and Phrasal Verbs as a source for that and also give a quote from Around The World In Eighty Days by Jules Verne as an example that points towards a meaning of postponing an idea (with the option of abandoning it completely):
Passepartout was not the man to let an idea go begging, and directed his steps towards the docks.
No, it means to give up on an idea that has been proposed or put forward, in favor, perhaps, of another, new idea. As for the old idea, it can seek acceptance elsewhere. In other words, it can go begging to be accepted by someone else, somewhere else!
It does not, by the way, necessarily mean a new idea came along, in which case the person who suggested the idea in the first place might be a little disappointed and say,
"So, what should we do with my idea? Should we simply let it go begging? I for one don't think we should."