There's an easy answer and a much more complicated answer.

The simple answer is that the Army itself refers to Private Jacob Parrot of the 33rd Ohio Voluntary Infantry in March 1863 as the 'first.' He was part of an April 1862 search and destroy raid on the Confederate rail system where they stole a train and tried to take out several bridges. Once the train ran out of fuel, though, he and the other surviving raiders got caught, but he was fortunate enough (especially given his location and the action committed) to be part of a prisoner exchange.

But...

Even the Army acknowledges the Navy had Congressional approval for their own version of the Medal of Honor two months earlier than the Army, and Navy records indicate multiple awards of it in 1862. Unlike the Army, the Navy hasn't put out an official recognition of their first recipient. The earliest with a confirmed citation is Quarter Gunner John Davis, who literally sat on top of a powder keg in the midst of a fire on the USS Valley City in February 1862 and was apparently awarded the Medal of Honor immediately afterwards.

However, that latter reference claims that he was the second rather than the first. Who was that? Well, another reference claims this may have been Robert Williams for non-specific conduct during an action in December 1862, but that date makes little sense as it's months later, I don't have access to that particular book to see how he supports that claim, and for a variety of reasons I'm actually a bit dubious about their methodology. To complicate it further, there were also several Medal of Honors awarded in 1863 for actions taken in 1861 in both services, so that puts yet another barrier to the 'true' first.

So why all the confusion? First, the record keeping for the awarding of the Medal of Honor in the Civil War was an utter mess, and a good portion of the citations attached to them have been lost. That gets to the more complicated answer, which is that the Medal of Honor from 1862 until 1916 was not the Medal of Honor of World War I and beyond. It was the only award available for large periods of that time and was bestowed for anything from actions that would be consistent with the vetting of it today to a battalion reenlisting, where since the Department of War didn't have proper records of who had reenlisted it ended up awarding it to the entire battalion. Would Parrot's or Davis' actions qualify under today's vetting for the award? Genuinely good question, and given the state of the records around it, even if someone were to do the work we will probably never know.

There have been a couple previous good answers on the origination that provide more details as to the mess the Medal of Honor entailed, with one by /u/ArmDoc explaining some of its genesis and a great series of detailed answers by /u/The_Alaskan on what finally was the last straw for the system, where a whole heap of Navy officers got theirs at Veracruz for more or less just being at the battle.

In fact, when you look at the service records of a bunch of admirals in WWII, you'll run to a number of Veracruz MoHs as somehow the Navy just never quite found the time to review the appropriateness of those awards. Or as one admiral not in that group once memorably put it about how officers had - and still do have - vastly lower thresholds to receive high grade awards compared to enlisted members, which was largely part of the underlying mess around the requirements for nomination for the MoH until Korea: "The Legion of Merit (about fourth or fifth in the order of precedence at the time) is more or less a Good Conduct medal for senior officers."

So in 1916, the requirements were tightened up significantly but even then not what they've (mostly) been since World War II - there were some in that war like Douglas MacArthur's that were fairly ridiculous - and of the first 10 or so "modern" MoHs awarded afterwards all went to the Navy in 1917-1918 for conduct that was generally heroic (saving ships and lives) but probably wouldn't qualify under today's standards. In fact, the most heroic action early in World War I was almost certainly Henry Johnson's crazy, gutsy stand in the Argonne in May 1918 for the Army - but being African American, it took until 2015 to finally receive recognition and award the MoH to his relatives.

However, in that first contemporary group of the "modern" MoH, the one that stands out was Edouard Izac in 1918. He had captured some important intelligence on German sub ops, been captured himself, could have spent the rest of the war in relative comfort (believe it or not, at that point in the war in some sectors POW officers were allowed parole on their honor on weekends to the local villages), and instead turned into a story that sounds a bit like James Bond if he were real and doing the Great Escape - sneaking out and somehow making it across half of Europe, risking his life multiple times, suffering a head injury that later contributed to lifelong issues, and getting the intel to high command as one of the few POWs to get out of captivity. It's been years since I read it, but the first modern retelling of the story - Dwight Messimer's Escape - from the Naval Institute was just straight out of a movie, and if some of the early MoHs from the modern era were reviewed, his would probably be the first that would have absolutely nobody questioning it.

Edit: cleaned up and clarified a few points.