"Double-odd" is a confusing term. Makes it sound like it's odd in two ways. A more common term for a number that's 2 mod 4 is "singly even" (because 2 only divides it once).
Let's just do the same we do with other classes of curves: Name them after the person who either came up with or popularized them. Thus, we might as well just call them Pornin curves (in line with Edwards curves and Montgomery curves and [short] Weierstrasse curves and Koblitz curves).
Edwards, Montgomery, and Weierstrasse are actually representations for elliptic curves (eg curve25519 is usually represented as a Montgomery curve, but a Edwards representation for the same curve is used in ed25519 ECDSA).
