How do you thematically include accidental findings ("serendipity") in a dissertation or paper? I'm asking for approaches to include interesting, but not perfectly fitting results in a dissertation or a paper.

During my PhD project I have made an accidental discovery, which is what I believe you call serendipity. The finding is related to the overall topic of my dissertation and certainly interesting, but it interrupts the "leitmotif" of my argumentation, as this discovery is really just the result of a stupid mistake. So my question here is: How do you eloquently include stupid mistakes (aka accidental findings) in a dissertation or a paper without sounding stupid or breaking the flow of arguments apart? Is there even a generalizing answer to this question?
“The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' but rather, 'hmm... that's funny...” (Isaac Asimov, thanks to EnergyNumbers for reminding me of it)



If you're worried that it will distract the flow of your thesis, why not put it in a “special” part of your thesis (e.g., an appendix) and refer to that from the main text, at the point that would be most logical.


  [Following the description of your experiment.] In the next few sections, I describe the results obtained from operating the Pocket Helium Flux Positron Annihilator on a variety of samples: metals (section II.B), graphene (section II.C) and heavy water (section II.D). You will also find in Appendix A a description of the observations made following an accidental operation of PHFPA without a helium flux [you may not want to be specific and say: some moron forgot to replace the bottle] which allowed to check what happens when electroneutrality is violated on the µm scale.


An appendix is a good place, or maybe a small section as the end of the relevant chapter.