I'm uncertain why we're answering the hypothetical from the point of an omniscient 3rd party. This would make more sense if it were presented as 3 points of view: the kid, the old lady, and the people the kid gave the dog to.
Treating all this as if it were the
claims of each of the parties involved rather than a gods-eye view of the entire situation, the old lady should be allowed to keep the dog.
The fact is that this is probably not a verbal contract because it A) appears to be too informal and unspecific to be a contract and B) involves no
consideration being put up by the people in exchange for the dog in the first place. On the other hand, if the agreement is really for them just to watch the dog for an unspecified time and then return it on demand, it still isn't a contract because now the kid is putting up no consideration; he's just getting a free dog-sitting favor.
As for what I'd do if I could simply decree a solution unbound by precedent, I'd confiscate the damn dog, auction it, and split the proceeds 3 ways between the kid, the old lady, and the people who gave her the dog.