Maybe you want to limit it to problems first proposed in the 20th century or later? Otherwise, proving the fundamental theorem of algebra took decades or longer. Gauss's first proof, for example, had a famous topological gap found in the 1920's.
Several of Hilbert's problems from 1900 are understandable at undergrad level and stayed open many years or are still open.
Then there are computer proofs, e.g. the 4-color theorem. Do those count? A few years ago we got a multi-year computer calculation finally proving that checkers is a theoretical draw, something everyone already knew even without an exhaustive proof.
Title of your question first sounded like merely "surprisingly difficult" rather than "stayed open at research level for a long time". One I like from some other MO thread: let C be a curve inside the unit square, connecting the lower left to upper right corner. Let D be a similar curve, connecting upper left to lower right. Prove that C and D intersect.
