From a UX perspective, should I make the width of my search suggestion drop list dynamic (width changes depending on the longest suggested term/phrase) OR constant (width remains the same as the width of the search box)?
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Every widget has a sweet spot that gives us what we need without being annoying. Search suggestions can be very similar for the first few characters, and the differences between the options happen at the end, or in the middle of the string. I argue that you need to find the proper width for this dropdown through experimentation, but I am biased towards going with a fixed size (big) for two reasons: white space and page rendering stability.
These are just two arguments in favor of a fixed width. I don't have any arguments in favor of the dropdown accommodating the longest result.
Also, love your username! :-)