Relationship friction, loss of control, ego-related conflict, power struggles, moral dilemmas and temptations, duty and responsibilities

Being pressured to change a relationship status by getting married, ending a friendship, etc.
A parent or grandparent pushing the character to have children
Family compelling the character to attend a particular school or pursue a career path
Being pressured financially (to make a purchase, save, invest in a sibling's venture, etc.)
Being expected to participate in social gatherings and attend religious services as a family
Being told to keep a secret or tell a lie to protect a family member
The family staging an intervention for the character
Being coerced into participating in an illegal family activity or helping cover one up
Loved ones advising the character to hide their true identity because they won't be accepted for it
Being pressured to live in a particular place (close to family or near family holdings)
Being compelled to continue the legacy of a family business (a ranch, the Mafia, etc.) 

The character breaking up with someone they like
Having to hide a relationship from family members
Only dating people who have been vetted and approved
Having to join a club or take up an activity because the family wants the character to
Applying to a school or for a job the character has no interest in
The character being surrounded by family (when they'd prefer space)
Always asking permission instead of being able to make independent decisions
Having to mentally prepare an argument for anything the family might take issue with
The character having to monitor what they say and to whom 
Always giving in to keep the peace
Having to attend therapy, counseling, or rehabilitation at the family's request