Thingie is what Tara Gilesbie refers to penises as. When Ebony is having sex with Draco next to his car after the Good Charlotte concert, she says "Then he put his thingie into my you-know-what and we did it for the first time."
No one is entirely sure if she had any real idea what a penis was, or if she was just trying to censor her story for younger audiences. Although seeing as she never censors "fuck," it was probably just Tara's ignorance of genitalia.
The "thingie" is referred to several times throughout the story.