Hard extendability problems
n-coloring the n x n Sudoku graph is trivial, but if some of the colors are given to you (the extendability version) it becomes NP-complete.

By the "Sudoku graph" I mean the natural graph whose associated coloring problem is Sudoku. Namely, suppose $n=k^2$ is a square. The graph will have $n^2$ vertices, which we will denote by $(r_1, r_2; c_1, c_2)$ for $r_1,r_2,c_1,c_2 \in [k] = [\sqrt{n}]$. For each fixed $(r_1,r_2)$, the vertices $(r_1, r_2; *, *)$ form an $n$-clique; for each fixed $(c_1, c_2)$ the vertices $(*, *; c_1, c_2)$ form an $n$-clique; and for each fixed $(r_1, c_1)$, the vertices $(r_1, *; c_1, *)$ form an $n$-clique.