Word tiling, where you must use each tile exactly once
I can't speak to whether it's a known problem, but it is NP-complete.

It's clearly in NP, so we just have to show NP-hardness. The following reduction is from the strongly NP-hard 3-partition problem, which asks, given a multiset of positive integers $[x_1,\ldots,x_{3m}]$, whether there exists a partition into $m$ submultisets such that the sum of each submultiset is equal to the integer $s=\frac{1}{m}\sum_{i=1}^{3m}x_i$. For each $i\in\{1,\ldots,3m\}$, make a string $a^{x_i}$. Make $m-1$ strings $b$. The target word $w$ is

$$a^sba^sb\cdots ba^s.$$