I've always thought it odd that the position of the Vice President exists. It's not a well defined position other than "back-up President." Probably the most important power they have to break a tie in the U.S. Senate. But I have an idea. What if the Vice President always had a vote in the Senate? So there would be 101 members of the Senate. 2 from each state, and the Vice President.
The only real difference is that the Vice-President would get to participate in the election of his successor when the election of the Vice-President goes to the House.
Personally, I think we should have at the moment 104 Senators, with the 101st to 104th Senators being Senators Carter, Bush, Clinton, and Bush Jr. by virtue of being ex-Presidents. (Presidents who resign or are impeached would not get that privilege.)