Family emergency
A family emergency was an urgent matter involving one's family.
In 2365, seeing Wesley Crusher's obvious distress, William T. Riker excused himself from a date, claiming to have a family emergency. (TNG: "Pen Pals")
In 2373, Annel, Kirayoshi O'Brien's regular babysitter, had a family emergency, leaving Miles O'Brien to tend to his own son in his wife Keiko's absence. Jake Sisko offered to fill in, much to Miles' relief. (DS9: "Business as Usual")