Symbolism, Imagery, Allegory
"Mayfly" is Evan's nickname for Cassie. And it's actually sweet—it doesn't mean that he thinks she's a weird little bug.
See, Cassie initially thinks of herself as being like a cockroach—a survivor who the Others are trying to stamp out. Evan prefers to think of her as the ephemeral mayfly, "here for a day and then gone" (36.112).
Evan's point is that human life is fleeting, but that's nothing new. He tells her:
"It's always been that way. We're here, and then we're gone, and it's not about the time we're here, but what we do with that time." (36.112)
Aww. Evan has a point. Life is fleeting, and we have to carpe that dang diem.
The mayfly, then, symbolizes the brief nature of human life. For Cassie, the nickname sticks… even though she doesn't quite get it. Maybe she would have preferred that her boyf referred to her as a cuter animal—a kitten? a bunny? even a mouse?—but she accepts the term of endearment "mayfly" because Evan, in his weird alien way, gave it to her.