2) Presidents can be remembered for good or bad reasons. Unfortunately it seems easier to forget those who did a decent job. So when ranking obscurity against performance, who are the undeservedly forgotten presidents?
1. The Most Obscure President, Relative to Time Since Leaving Office
Results here are for presidents through Reagan. Here I assume obscure = least remembered = least discussed in print at a certain recent point in time. (We don't care that Zachary Taylor was discussed in 1849, of course he was, he was in office!) So I looked at Google Ngram mentions of the president's names in the year 2000. This means I excluded all presidents who did not complete their administration by that point (Obama, George W. Bush, and Clinton) and also excluded George H.W. Bush because of confusion with his son in the 2000 primary season. I included common variants of their names ("Chester A. Arthur" and "Chester Alan Arthur"; "JFK" especially may be overinflated by mentions of the airport.) I did not include nicknames (Tippecanoe, Ike, etc.) I consider Cleveland to have left office in 1897 (no special treatment due to non-consecutiveness.)
Once I had the number of Google Ngram mentions per president, I compared against how long they'd been out of office. Not surprisingly, the trend is that the longer you're out of office, the more obscure you are. For all of these graphs, Y-axis is remembrance (%Ngram mentionsx10,000), and X-axis is years out of office. (X-axis might be counter-intuitive; earlier presidents are on the RIGHT.)
It's reasonable to think that most of the forgetting occurs in the first couple decades; that is, from 1 year after they leave office to 21 years after they leave office, people will forget faster than from 101 years to 121 years. And indeed that's the case.
Above you can see the curve for (more recent) 20th century presidents, which is three times steeper (we forget the more recent ones faster) than the overall curve for all presidents. Also, the older group starting with John Quincy Adams through Cleveland is essentially flat. So, after they've been gone a century, we've basically forgotten whatever we're going to forget. (I started with John Quincy Adams to avoid inflation by Founding Fathers; those presidents were already well-known for their involvement with the Revolution and Constitution, and I want to learn about remembrance relating to what they did as president.)
Now we can look for presidents that are furthest off-trend from the curve. I modified the original curve from linear to a power function, which reflects the early forgetting trend better than a linear function.
Sure enough, relative to how long he's been out of office, Gerald Ford is the most obscure. He is furthest below the curve, on the lower left. Most presidents are in a sea of average-to-obscure just below the curve, starting with Coolidge as we go back. But there are plenty of more memorable outliers.
So who are the best-known, relative to the the time since they left office? FDR, LBJ, JFK (remember, airport signal), Wilson, Teddy Roosevelt, Lincoln, Jefferson and Washington. Honorable mentions are John Adams, James Madison and Andrew Jackson. What's interesting is that being famous before you're president (usually for being a war hero, but in one case being an actor) doesn't seem to matter for how well you're remembered, even if it helps you get elected. (Today do we talk about Grant the president, or Grant the general?)
2. How Are Presidents Remembered Relative to Performance? Who is Most Under-Rated?
My source for presidential performance is the Wikipedia article on ranking of the presidents; I averaged every survey they have in the scholar survey results. I convert the average of all the scores into a decimal between 0 to 1. 0 means everyone unanimously agreed he was the best president, 1 that he was terrible. These surveys span 1948 to 2015. The curve in the previous graph predicts how well-remembered they should be based on how long they've been out of office. For each, their distance above or below the curve is the relative remembrance. This compares the relative remembrance to their historical ranking. (For grins, I compared my own ranking against the historians and got an R^2 for 0.4527; there was general agreement except relative to historians, I really don't like John Adams and LBJ.)
As it turns out, there may be justice after all. The better the president, the more likely they are to be remembered. (Although it must be said, if the winners write history, this is also what we should expect to see.)
Looking at the two "unjust" quadrants (bad but remembered, or good but forgotten), let's focus on the good-but-forgotten; they're the ones we should be thinking more about, plus the good-but-forgotten stand out above the curve much more than the bad-but-remembered.
I circled the three most unjustly forgotten presidents. As I'd expected, Polk was a good but forgotten president (come on, click on that and read about him, he deserves it!) In particular, he explicitly set a number of ambitious foreign policy and domestic goals, and achieved them, prosecuting the Mexican War and expanding to the Pacific, settling the Oregon Territory question peacefully but to American advantage even against Victorian Britain, and establishing the forerunner to our modern treasury. (He would have been a very good candidate for futarchists!) Eisenhower and Truman are also standouts underrated by the attention we pay them.