I don't know about reliable sources for it (I could find some, most likely, but eh, I'll just put this first and then maybe go looking), but I can say that "haven't the slightest idea" is a phrase in common use, to the point where you can often find even that shortened to just "haven't the slightest" or similar. I like "haven't the foggiest" best, myself.
"Haven't had" is a different thing altogether. To say you "haven't had the slightest idea" is to say you did not, in the past, have the slightest idea, whereas "haven't the slightest idea" is saying that you currently, in the present, do not have the slightest idea.
"Don't have" may be more technically correct, perhaps, but "haven't the slightest" should be acceptable as well.
And now I'll go looking for sources.
Edit: from what I can tell from very brief searching around, either form is more or less acceptable, though the "haven't the"/"haven't a" form is apparently more Euro-popular than American. Or something. What can I say, I'm terrible at research (which is why I never got into teaching/journalism! ...among other reasons).