I had a book of parlor games from that era. Lots of parlor games. I remember one called "kiss the candlestick" in which a girl was blindfolded and given a candle and complex directions but somehow it ends up it was she who was kissed. (go figger)
Music. A piano (or a melodeon) in a middle-class household. (Guitar or fiddle or banjo in a poorer household. Maybe home-made percussion as well in the latter case.) People might take turns playing, singing. Again, depends on class, but you'd get art songs (think, Schubert's) or popular songs (in that era "After the Ball" I think was out. "
Because" "Bird in a Gilded Cage" "My Wild Irish Rose"). ETA: okay, if they are rich, a child might want to sing "Bird in a Gilded Cage" but a parent in the corner doing needlework might protest it: "Do that lovely Schubert instead." Because this lyric would have been considered indecent:
And her beauty was sold,
For an old man's gold,
She's a bird in a gilded cage.
Story telling. Taffy pulls and popping corn.
ETA: stereographs. Maybe even early phonographs for rich people.
badly formatted, but here's a lot of diversions for children through young adults:
http://www.archive.org/stream/kindergartenpain00clin/kindergartenpain00clin_djvu.txt
Recitations. Used to be a big thing at school and at home. They'd do an act of Hamlet, or recite Spartacus's speech to the gladiators. Look for pdfs of "the MacGuffey Reader" or "The Comprehensive speaker" at googlebooks for examples. The guys, to show off to the girls, might pick something with swordplay in it.
ETA to add: Hey, I'm old, but I'm not this old, okay?