Out of curiosity who was the first member of each House to become a father while in office? What about the Congress of the Confederation and the Continental Congress?
If we limit it to the United State Congress, it probably was Senator Rufus King of New York. He was a member of the 1st United States Congress which convened on March 4, 1789. King’s son Charles was born twelve days later on March 16.
Update: It wasn’t King. While the first Congress convened on March 4, King wasn’t a part of it. The New York state legislature (which chose Senators back then) didn’t get around to naming anyone until July 16 and King didn’t take office until July 25.