To steal in a sophisticated and cunning manner. To convince (much like a confidence man) someone to part with a valuable and, if the finneagler is particularly skillful, to do so believing that it is actually what the finneagled want. Furthermore, to finneagle would be to convince someone that by handing over their valuables, it is helping the finneagler out of a jam or trouble.
I wonder if I could finneagle some cash. Who would be a pushover....who?