As popularized by Milhouse on The Simpsons, a phrase referring to the father of a child who's parents are divorced and split custody of the child.
In such situations, what can happen is the mother has custody of the child during the week, while the father gets to spend one or both days of the weekend with him or her.
Phrase is especially true if the mother is remarried or has a boyfriend, hence the child's father figure during the week. This makes the biological father the weekend dad.
Kid- "Where are we going this Saturday weekend Dad?"
Father- "Stop calling me that! We're going to the water park."
Kid- "Thanks, weekend dad!"
A politically correct rewording. The original term was/is "Weakened Dad" which is loosely based on a spouse or former spouse's actions toward her child's father.
"While they were married, Susie did not feel that her husband made a large enough salary, so she turned him into a Weekend Dad"