Hey pothios, I'm a vocal coach myself, and yes I normally get my student to sing me a song on the first session. The songs range from "Doe a deer a female deer...", to the alphabet song, to 'When You Say Nothing At All' to something Mariah Carey might sing.
I do it just to gauge a student's pitching/rhythm and general ability in music, so it doesn't really matter which song is sung. You can sing "London bridge is falling down..." and it'd be fine for me and I'm sure for many other teachers as well. So there's nothing to worry about.
Like someone before me said, you're there to make mistakes, and teachers are prepared to see you make mistakes and do something totally funny and humiliating. But that's how everyone learns, so no nothing to specially prepare. Just anything that comes to your mind, even nursery rhymes!
91880920,
Gary.