Not really to do with partial evaluation or the unrelated idea of a partial function.
A key motivating (non-Haskell) example behind continuations is the idea of replacing "return" with a function call. This is continuation passing style, and obviously when you call a subroutine in CPS, you need to give it a function to call when it completes: a "continuation" which is contrived to be equivalent to what would happen when that particular subroutine "returned" in normal direct style
