"Biddy" is actually a very interesting word because it has two separate origins, both fairly well-documented, which is unusual for a slang term. The primary meaning of "biddy" is "chicken," and it first appeared in the early 17th century. The word probably came from the nonsense syllables used to call chickens -- something like "here biddybiddybiddy," I suppose. By the late 18th century "biddy" had been adopted as a derogatory slang term for women, much in the same unfortunate way that "chick" was in the 1960's.
However, "biddy" in this sense might have died a welcome death had it not been for the influx of Irish immigrants into the U.S. in the early 19th century. Young Irish women often had their passage paid by upper-class American families, for whom they would then work as domestic servants while they paid off their debt. The practice was so widespread that such women came to be known as "Biddies," from a shortening of "Bridget," a common Irish women's name. This use of "biddie" reinvigorated the word, and ever since it has been employed by insolent children to torment their elders.
Sometime in the last 20 years or so, the term was recycled to mean a woman of any age. The connotation may exemplify the earliest expression synonymous with the concept now known as MILF.
"The old biddy is playing bridge"
"I'm trying to pick up a biddy"