How to gain influence:
Assign your family members as Governors. While I don't see it noted anywhere that governorships grant influence, it slowly creeps up while they are in office. It also grants the character XP, levels, traits, and households, all useful things.
Have a family member lead active offensive armies. I usually reserve my main attack force for my faction leader as this role tends to accrue the most Influence the fastest, assuming you are an active attacker.
Assign household items that passively grant Influence to family members.
Accept, not seek, wives that grant influence.