Failures and mistakes, relationship friction, duty and responsibility, moral dilemmas and temptation, losing an advantage, ego-related conflict, miscellaneous challenges

A parent's extra-marital affair is uncovered
Discovering the existence of half siblings (say at the reading of a will or when someone shows up at the door claiming to be related)
Finding out a crime was covered up (someone was paid off, a family member was “sent away to school” to avoid consequences, etc.)
A family member is outed for drug abuse, alcoholism, or a gambling habit
Discovering a family member has a fetish or unconventional sexual preferences
A hidden pregnancy
Discovering one's ancestors were war criminals, racists, or supported such things
Discovering ties to the occult
Finding out one is adopted (or a sibling is)
A discovery that the family's wealth, property, or power was obtained illegally or through immoral means
The truth coming out about a family business (that it's almost bankrupt, that it was won in a poker game, that it was built on deceit or through the hard work of others, etc.)
The discovery of a forced marriage
Uncovering the source of a feud
Family abuse coming out into the open (physical, emotional, or sexual)
Finding out that a mental condition or genetic physical ailment runs in the family
Finding out an ability runs in the family (psychic sensitivities, a gift that has always been suppressed or hidden for safety, etc.)

Strained relationships
Awkwardness now that another's secret is out in the open
Family members taking sides
Family members pressuring others to let it go to keep the status quo
Needing someone to talk to but having no one due to broken trust
Unwanted publicity if word gets out
Guilt trips to stay quiet
The burden of knowledge erasing their innocence and changing how they view family