The short "y" represents the sound of "i" before "r" in the words zephyr, martyr, satyr. You can see "y" representing the value of short "i" in other words like analysis, crypt, etymology, Egypt, myth, oxygen, pterodactyl, system etc. derived from Greek but borrowed through Latin or French.
The common pattern in zephyr, martyr, satyr can be traced back to their ultimate Ancient Greek origin which has a similar ending and and the letter upsilon (υ).
The relevant part from the etymology of the words from OED:
- zephyr: < classical Latin Zephyrus < ancient Greek Ζέϕυρος
- martyr: < post-classical Latin martyr < Hellenistic Greek μάρτυρ
- satyr: Partly < classical Latin satyrus and partly (in later use) < Middle French satire, satyre (French satyre) (late 14th cent.) < classical Latin satyrus < Ancient Greek σάτυρος
The letter “y” was a readoption in Latin of the Greek upsilon (υ), and was used for Greek borrowings.