The original mechanism was for each elector to cast two votes for separate candidates (at least one of whom needed to be from outside the elector's state). The top vote-getter would be President, and the runner-up would be Vice President.
This never really worked as intended. In the first two Presidential elections, it mostly worked, but only because there was a strong consensus in favor of Washington for President: every elector cast their first vote for Washington, making the second vote effectively a vote in a contested election for Vice President. The 1788-9 election had a wide open field with no organized campaigns, with Adams winning with a strong plurality (just short of a majority). The 1792 election was a two-way partisan contest between Adams (backed by the Federalists) and Clinton (backed by the Democratic-Republicans).
After Washington retired, the parties contested both seats, which is what wound up breaking the system. Each party ran two candidates for President, with the understanding that one was to be President and the other to be Vice President. The plan was for 1-2 electors to cast their second vote for someone else instead of the VP candidate, ensuring the top-of-ticket candidate would win. In both elections where this was attempted (1796 and 1800), the parties botched the job.
In 1796, too many Federalist electors cast their second vote for a different candidate, and the D-R candidate (Jefferson) ended up getting the second-most votes. And having a President and VP of rival parties turned out to be a bit of an undesirable mess.
In 1800, the two D-R candidates (Jefferson and Burr) wound up tied with each other, leaving the lame duck Federalist-majority House of Representatives to break the tie. A large faction within the House tried to break the tie in favor of Burr, producing a prolonged deadlock, during which Jefferson threatened civil war if Burr were elected.
If you want to preserve the runner-up system, I think one of the following needs to happen:
- Avert the emergence of a two-party system. This is very difficult, since Presidential systems lend themselves to an administration vs opposition party dynamic, and since two-party politics was already firmly rooted in the Anglo-American political tradition.
- Make the parties get the coordination job right in 1796 and 1800. This would preserve the constitutional mechanism, but subvert the practical application into something still very close to the modern mechanism of separate ballots for the two offices.
- Adopt a different runner-up system that's too hard for the parties to try to subvert. For example, something like Borda Count might work (each elector's first vote counts for 2, while their second vote counts for 1, so a party would need a 2/3 majority of electors to capture a majority in both slots). Or just have each elector cast a single vote, but that could break down in the first election if Washington were to win unanimously, leaving no runner-up.