half a notion
{n. phr.}, {informal} A wish or plan that you have not yet decided to act on; a thought of possibly doing something. - Used after "have" or "with" and before "to" and an infinitive.
I have half a mind to stop studying and walk over to the brook.

Jerry went home with half a mind to telephone Betty.

[TahlilGaran] English Idioms Dictionary ▲