、。
Yesterday, through some strange chance, I happened to see a copy of my father's family register.
Literal
Yesterday, strange thing [by-で] father [genitive-の] family-register-extract [genitive-の] copy [object-を] saw [end-up-てしまいました].
ひょんなことで is a variant of the more common ひょんなことから, both meaning 'through some strange chance.' ひょんなことから emphasizes the origin ('starting from some strange thing') while ひょんなことで emphasizes the circumstance ('in some strange way'). 戸籍抄本 ('family register extract') is a very specific piece of Japanese bureaucratic vocabulary: 戸籍 is the koseki, Japan's family registration system, and 抄本 is an 'abstract' or 'extract' — a partial copy of the full register. ~てしまいました adds the 'ended up' nuance of ~てしまう, marking the viewing as unintended or regretted (or surprising).