To make an effort with someone means to put energy (usually emotional) into a relationship. The relationship is usually a family relation, a romantic relation, or a friendship.
"I make an effort to keep a strong relationship with my parents."
"My boyfriend bought me flowers as a gift, he really makes an effort."
"My friend always forgets my birthday, she should make more of an effort."
It's a phrase that is only usually used in casual speech.
But sometimes you might hear it in a work environment :
"Susan is always late for her deadlines. She should really make more of an effort, or she could get fired"
Does that make sense? :)