What contributions from your advisor must you specifically acknowledge in your dissertation? When you're writing your dissertation, your advisor will undoubtedly make many contributions. They might suggest literature to look into or ways of tackling a problem. Many of these contributions I assume would be acknowledged sufficiently in the acknowledgements section where you thank those who helped, your advisor above all.

But what about when you're discussing something specific, maybe trying to figure out some problem. When you're working through it together, your advisor says "What about you try X?". It turns out that X solves the problem.

In the section of my dissertation where I discuss the problem and present my implementation of solution X, should I include something like a footnote stating explicitly that X was suggested to me by my advisor in conversation? I'm interested more in the issue of academic honesty/integrity than issues surrounding my relationship with my advisor (e.g., whether it might tick him off to not be specifically acknowledged in the section on X).
It really depends on how original the contribution is.  At one extreme, your advisor might say something like "Hey, you're integrating an oscillating function times a smooth function.  Have you tried integrating by parts?"  This sounds impressive to a beginner and pointing it out could be a valuable contribution, but your advisor is just cluing you in to something every professional knows.  There's no need to offer explicit credit.

At the other extreme, your advisor might hand you a sheaf of handwritten notes and say "Here's what I tried and where I got stuck when I thought about this problem a few years ago."  If you make use of original ideas from the notes, then you should certainly attribute them to your advisor.

The tricky part is how to gauge originality.  You might not have been able to tell at the time, but looking back you'll often have a better feeling for this.  If you recognize an idea as being relatively standard, then offering general thanks in the acknowledgments should be enough.  If the idea still impresses you as you are writing your dissertation, then it deserves attribution.  If you aren't sure, you can always ask your advisor.  For example, "One of the decisive moments in my studies was when you suggested I should use Bayesian hierarchical modeling.  I haven't seen other papers using this approach, so I feel I should explicitly credit you with the idea in my dissertation.  Does that seem appropriate?"  Your advisor will either gracefully accept credit, declare that it's not worth worrying about, or offer background references you've missed.