Lina is Lefty and Desdemona's cousin who lives in America; she's also the mother of Tessie, Cal's mom. Once Tessie is born, though, Lina's character kind of fades into the background.
Lefty and Desdemona move in with Lina and her husband, Jimmy, when they arrive in America, and Lina keeps the secret that her married cousins are actually brother and sister. She's good with secrets because she has plenty of her own. Even though she's married to a man, she actually likes women. Being a lesbian isn't exactly acceptable in Greek culture or in the 1920s, so Lina's basically navigating a secrecy double-whammy. She says, "People think if you like girls, you like every single one. I was always picky. And there wasn't much to pick from" (2.1.59).
Jimmy is a piece of work. He's described as an "amateur herbalist; antisuffragist; big-game hunter; ex-con; drug pusher; teetotaler" (2.1.76). Teetotaler, eh? Considering he also smuggles liquor from Canada, that's a little ironic. He ends up dying trying to make a deal by crossing a frozen lake in a car.
Lina is kind of glad when Jimmy Zizmo is gone, though she dutifully wails histrionically at his wake and mourns for the requisite 40 days (before promptly tossing away the black). "Forty days is enough," (2.3.9) she says, but we think she's just giving the people what they want. Now she's ready to live her own life.
We don't think Lina finds out, but Jimmy Zizmo's death is faked and he reemerges as the Prophet at the head of the Nation of Islam. He goes by many names; he goes by no name. He's a little crazy about some things, and kind of right about others. That's true for everyone though, right? Where do we start our own cult?