Interesting topic, it depends on how you define "worst". "Worst" as in the president whose ideas policies and administrative decisions I disagree with the most, "worst" as in politically ineffectual (regardless of the actual merit of the proposed policies), "worst" as in not vindicated by history, or some people take it to mean overrated, which is really not the same thing but it is related idea.
You could argue, for example, William Henry Harrison is the single worst president because he accomplished nothing to speak of, but it's because he died within a month of taking office, but is that really fair? Well again it depends on how you define "worst".
For my pick, Woodrow Wilson is probably the single worst President the United States ever had. Many far more educated than me have presented the argument better than I can, this is one of my favorites:
I would argue FDR is the most overrated president and also just a bad president for establishing the precedent of greatly expanded federal spending and federal power. To me though one of the most bitter pills with FDR is his agenda was simply the same agenda as Hoover's (seriously Hoover was doing the New Deal before there was a New Deal), but he managed to twist it to make Hoover sound like a villain and portray himself as the hero. It's also amazing that he extended the Great Depression single handedly and brought unemployment to record heights and God knows how many people he starved to death and how much sheer human misery he created under the guise of "doing something" and yet gets absolutely no blame, in fact people credit him with "saving" the United States economy. It's amazing.
I don't think he's necessarily the worst from the perspective of when you have a true existential threat to the country, it does pay, from a purely utilitarian perspective, to have a shifty, slick son of a gun in charge, someone who will keep the enterprise on the rails no matter what morally repugnant things he has to do. In other words you can admire his ability to lead and still think he's the devil himself. The fact was, FDR knew Hitler had to be stopped cold long before most of the US was willing to get involved and he did everything he could to support the Allies before Pearl Harbor happened. It takes an evil man to recognize the ambition of another evil man sometimes.
However, I would argue that FDR's service, while effective, was a Pyrrhic victory for the United States as an entity because his terrible ideas about how government should run have become the de facto political ideology of our day, he has shifted the conversation so far toward the radical left you can't even discuss mundane ideas like allowing citizens to opt out of social programs at their own discretion without being branded some kind of radical pariah.
Seriously tell someone you'd rather invest in stocks, bonds and/or real estate than give money to Social Security because it's a better way to secure your retirement and see the funny looks you get, and that's because of FDR normalizing the idea the government has to expand endlessly and offer us every conceivable service or we'll just all die. All I can say is Thank God his court packing plan never worked out or we'd really be in some poop now. So FDR is my second worst president, because while his legacy is every bit as destructive as Wilson's, he was at least effective at keeping the country cohesive during a crisis and I can admire the incredible skill with which he crafted the narrative and established himself as one of history's "good guys".
I noticed some people ragging on Lincoln. While I agree Lincoln is lionized and treated as this saintly figure (a reputation he does not deserve) and in fact has a lot of evil to answer for, I wouldn't call him the worst president by a long shot. Yes, Lincoln suspended several Constitutional rights and there's a lot to be argued for him conducting the war poorly (both in the sense of it took the Union a ridiculously long time to win it due to horrible mismanagement and a lack of competent leadership, after all McClellan did not appoint himself nor was he relieved of duty timely for example, and in the fact that civilians were abused, Licoln could have reigned in Grant, Sheridan and Sherman and did not).
Additionally, he's also not the racial equality advocate or opponent of slavery he's made out to be, not even close. He did detest slavery and thought it shouldn't be, but he really did not want to get rid of it until it was politically advantageous for him to do so. He was as racist as everyone else in his day and several verified quotes of his confirm he did not think a black man could ever be intellectually as potent as a white man. In fact, he actually sponsored an expedition to exile the freed blacks to a colony and only abandoned the effort when the attempt failed utterly.
The point is, if you don't think Lincoln was a good man, I can see where you're coming from. However I wouldn't argue he's the worst for two main reasons:
Lincoln may not have been a good person, but like FDR he was an amazingly adept politician. One part of the Lincoln mythos that is actually true is he really did come from nothing and progressed to the highest rung of society through sheer talent and tenacity despite many obstacles and failures. Remember, intelligence and character are not the same thing, professional accomplishment and being a good person are not the same thing either, although we tend to culturally conflate them sometimes. Due to his origins he was able to make others thing he was a stupid man, which he used to pull several political coups and upsets. Lincoln also carefully cultivated his image. For example he'd tell Northern audiences and Southern audiences different things on the same topic, which was far more effective back then because you didn't have the ability to search video archives. He also may have lied about who he was and his personal beliefs (for example he played up his reputation as a rail splitter and hard laborer over being an attorney who negotiated right of ways, and he made many overtures to religious references while possibly actually being an atheist). Yes this all makes him a liar but it makes him a good liar.
Okay, why is this redeeming? Because at that moment in time, the United States (as in the US at the time which lacked the Southern states) was incredibly divided on the war and any number of other issues, and it needed someone cagey and cunning to just keep it politically cohesive and functional at all. The fact Lincoln managed to do this, and win over just enough support from factions within the USA that opposed him directly by extending just enough olive branches to win over the moderates just enough to maintain power, is actually quite impressive. If you view the president's job as keeping the country intact and the government being perceived as legitimate despite no one actually liking the POTUS or the government, Lincoln did something exceptional. Whether we like it or not, the USA needed a shifty, slick son of a gun who could put all morals aside and get the job done in the driver's seat, and we had it in Lincoln. You can still hate his guts as a man and acknowledge what he did was damn effective. Lincoln didn't really have solid morals imho, but he was tremendously practical. He got stuff done.
Additionally, if Lincoln had remained in power another term, the South re-entering the Union probably would have gone a lot smoother. Andrew Johnson was from a political faction that wanted to really damage and punish the South for the secession, Lincoln wanted to console and mend fences and get the South up and running again economically. It's not because he was good and compassionate, it was because Lincoln wanted the federal tax revenue. Nevertheless the fact remains Lincoln would have been much softer and even handed in restoring the southern states and set a very different tone from the punitive Reconstruction effort that actually came to pass. I will be so bold as to say if Lincoln hadn't been assassinated, the Lost Cause would never have come to such prominence and while there'd still be Confederate memorials I think they'd be very different in character and focus, more focused on memoralizing the losses and tragedies of the war rather than trying to edify its heroes. On the downside I think the Jim Crow era would probably have lasted longer, Lincoln was not particularly interested in helping the freed blacks achieve anything because it was politically volatile and not conducive to his goals.
This last paragraph is why I would in fact argue that Andrew Johnson is actually a much worse president than Lincoln, he didn't have Lincoln's skill, was also a son of a bitch, and he set the tone for tension between the reconquered South and the rest of the country for decades to come. He didn't do it with something spectacular like a war, but he imho did far more lasting damage to America and we're still dealing with his poor decisions. At least Lincoln's evil stayed (mostly) in his own time.
In fact I don't think Lincoln even rates the same as an FDR, Lincoln also provided good crisis leadership, but his legacy wasn't nearly as toxic.