How do you thematically include accidental findings ("serendipity") in a dissertation or paper?
“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.