MOLLY. English and Irish: long-established pet name for MARY, representing an altered version of the earlier pet form MALLY.

POLLY. Variant of Molly, now established as a given name in its own right. The reason for the interchange of M- and P- is not clear; similar to Peggy for Margaret.

Source: A Dictionary of First Names by Patrick Hanks and Flavia Hodges, Oxford University Press, New York, 1991, Pages 240, 269.