Should I cite my supervisor in a thesis?

I think it is common and well-known that the supervisor sometimes helps the PhD student. In fact, the main work of the supervisor is giving ideas that they come up with due to their experience and knowledge. In many cases, this ideas will not work, in some cases they may. It is up to the PhD student to work out the details and see what comes out in the end.
Thus, I don't think that you should cite your supervisor at every idea she came up with. It should of course be clear in the document who supervised you and you might want to add an acknowledgement.

Two small points to finish:

• If the result is important enough for a stand alone publication, i.e. if you are writing a paper out of it, your supervisor deserves to be one of the authors (then again, this should be rather common for most papers written by a PhD student...).
• If you are in doubt, ask your supervisor about it. She will know how to write it and you can be sure that she agrees with your choice.

Since this is a concrete item (rather than some generic advice) and someone else (meaning "not you") came up with it, you need to cite it.

A personal communication reference would be in order. Your first text sample seems fine to me though (once the reference is added): I assume that you do share the suspicion of your supervisor, so the text is not misleading.

Asking your supervisor about this is definitely a good idea, though.

Many years after my PhD thesis with Andy Gleason I wrote this in a tribute:

The central mechanism for producing examples and counterexamples showing the theorem was sharp came later that spring. Moreover, I think the idea was his, although I didn’t give him due credit then. (http://www.ams.org/notices/200910/rtx091001236p.pdf)

He didn't say a word when he read the thesis.

I don't think you need to thank her for this particular idea. But you should ask.