What's the difference between on my mind and in my mind?

"On" dá a ideia de em cima, sobre algo. "In" dá a ideia de dentro de algo. Então o correto seria "I have something IN my mind".

Na verdade, "on my mind" é correto, e mais comum do que "in my mind" em minha experiência. Trata-se de um caso específico, que foge à regra padrão do "in/on".

"I've had this song on my mind for almost a week now."
"This song has been on my mind since the first time I heard it."
"You're always on my mind." (Música)

Por outro lado, você pode dizer "to have something in mind" (sem o "the") no mesmo sentido de "ter algo em mente" (como um plano, ideia, linha de ação, etc.)

"There isn't much to do here. Do you have anything in mind?"
"Do you have anyone in mind for the job?"

Neste caso, "on mind" não seria apropriado.

When something or someone is on your mind, it means that you think and worry about this thing or person a lot.

When you have something or someone in your mind, you just think of this thing or person without the worrying connotation.

So the song is in your mind, and it just can't get out! damn!