Crossword puzzles flex one very specific piece of cognition--the ability to find words, which is also known as fluency. Fluency is a type of process based in the speech and language centers of the brain. So doing crosswords might help you get better at word finding, but that's the sum total of their positive benefits to your brain.
misnomer. n. 1. an incorrect or unsuitable name or term for a person or thing. 2. the act of referring to a person by the wrong name.
In non-technical contexts, the term homonym may be used (somewhat confusingly) to refer to words that are either homographs or homophones.