It is the American version of British All-Hallows summer, French été de la Saint-Martin (feast day Nov. 11), etc. Also colloquial was St. Luke's summer (or little summer), period of warm weather occurring about St. Luke's day (Oct. 18). An older and simpler name for it was autumn-spring (1630s).
masc. proper name, 12c., from Old English Petrus (genitive Pet(e)res, dative Pet(e)re), from Latin Petrus, from Greek Petros, literally "stone, rock" (see petrous), a translation of Syriac kefa "stone" (Latinized as Cephas), the nickname Jesus gave to apostle Simon Bar-Jona (Matthew xvi.17), historically known as St. Peter, and consequently a popular name among Christians (Italian Pietro, Spanish and Portuguese Pedro, Old French Pierres, French Pierre, etc.). As slang for "penis," attested from 1902, probably from identity of first syllable.
The common form of this very common name in medieval England was Peres (Anglo-French Piers), hence surnames Pierce, Pearson, etc. Among the diminutive forms were Parkin and Perkin.
To rob Peter to pay Paul (1510s, attested in slightly different wordings from late 14c.) might be a reference to the many churches dedicated to those two saints, and have sprung from the fairly common practice of building or enriching one church with the ruins or revenues of another. But the alliterative pairing of the two names is attested from c. 1400 with no obvious connection to the saints:
Sum medicyne is for peter þat is not good for poul, for þe diuersite of complexioun. [Lanfranc's "Chirurgia Magna," English translation, c. 1400]