When were US Presidential candidates' names first shown on the ballot?

It looks like it might have first happened in the election of 1804. However, this answer may not mean quite what you think it means.

One problem with answering simple questions about the quadrennial US Presidential election, is that effectively there is no such thing. Its kind of a polite fiction. Instead, there are 51 separate elections that (mostly by tradition) are all held on the same day. Each state runs its own election, by its own rules. These elections technically only appoint "electors" to the real election.

While the Constitution is fairly explicit on how things work from the Electoral College onward, the limits the US Constitution places on these state elections are surprisingly minimal. If a state wishes to simply have its governor or state legislature appoint the electors and have no popular vote whatsoever, that's their right. Of course that would be cartoonishly undemocratic, so no state has done that since the end of the Civil War. But the point is, the rules are entirely up to the states, and as long as the states follow their own rules (and don't do anything unconstitutional with any vote they do have), nobody outside that state really has a say in what they are.

The original idea was that, rather than a direct election, states would send their wisest men to the Electoral College. This was pictured as a body similar to the Vatican's College of Cardinals: They would get together in a proverbial smoke-filled room¹ and decide who would make the best president. Every elector got 2 votes, and the top 2 choices, if they got more than ¼ of the votes and didn't tie, became POTUS and VPOTUS. There are rules after that for how the votes fall back into Congress, and it was kind of expected that would happen a lot, due to there being lots of regional candidates and none of those evil nationwide political parties like the British Parliament was afflicted with.

The system worked mostly as expected for the first couple of elections, because George Washington was so popular there wasn't much thought of selecting anyone else. For both elections, only Maryland and Pennsylvania had popular votes for electors. It seems like the choice for voters there was how "Federalist" their elector was, which would perhaps impact their Vice Presidential choice.

However, we can see with the very next election that it wasn't working anymore. There were in fact political parties, and faithfully following the system meant the VP was from a competing party, which has all sorts of obvious practical problems. The solution to this was to make sure to send electors who would promise to spend both votes on the preferred party's two candidates. This exposed a "bug" where if at least one elector didn't vote for a different VP candidate, the top 2 candidates tied², sending the election to Congress.

That bug got patched with the 12th Amendment, which made VP and POTUS different votes in the Electoral College. Arguably, this normalized the behavior of electors being rubber stamps for particular parties, by incorporating that process into the Constitution. The first election run under this new system was the election of 1804.

Of course the electors as designed were supposed to be exercising independent judgement, and there appears to have been a bit of resistance to just flat out replacing their names on the ballots (in states that had ballots).

Supreme Court Justice Storey in 1833 summarized the existing situation thusly:

It is notorious, that the electors are now chosen wholly with reference to particular candidates, and are silently pledged to vote for them. Nay, upon some occasions the electors publicly pledge themselves to vote for a particular person; and thus, in effect, the whole foundation of the system, so elaborately constructed, is subverted. The candidates for the presidency are selected and announced in each state long before the election; and an ardent canvass is maintained in the newspapers, in party meetings, and in the state legislatures, to secure votes for the favourite candidate, and to defeat his opponents. Nay, the state legislatures often become the nominating body, acting in their official capacities, and recommending by solemn resolves their own candidate to the other states. So, that nothing is left to the electors after their choice, but to register votes, which are already pledged; and an exercise of an independent judgment would be treated, as a political usurpation, dishonourable to the individual, and a fraud upon his constituents.

As for what was on the actual ballots, I'll do my best here. It looks like the 1796 Maryland ballot just had Elector names. Here's an image I dug up of a tally for one district:

The next 2 elections for that same district also tallied by elector names, but the 1804 election did not:

New Jersey that year put both pieces of information on their returns:

Pennsylvania's return for that year is organized in the same way. All the other states that had a popular election and an archived copy of the actual results listed only by elector name. Of course there were only 8 states with a popular vote that year.

I then back-poked through the previous elector ballot records of all the states that had them, thanks to a really handy website at Tufts, and most every state had electors named on their archived actual archived election results³. However, there were a few states that they didn't have archived hardcopies to look at, and of course many states were still not having a popular election at all.

So its conceivable some state may have done it earlier, and we just don't have a record for it. 1804 is just the earliest I can find doing it for sure. Based on that also being the first election year after the Constitution was changed to accommodate mass use of pledged electors, 1804 seems like a good touchtone year.

^{1 - Technically 51 smoke-filled rooms; one for each state.}

^{2 - In a way, it was this bug that ultimately led to the death of Alexander Hamilton.}

^{3 - Endearingly, many appear to have been hand-written, in cursive}