So says this study from 2008.
The article mentions that children who study an instrument score higher in verbal ability and pattern recognition, as well as finger dexterity and auditory discrimination. The authors of the article suggest that the former two are not related to music.
I think it's fairly obvious that pattern recognition is intrinsic to musical study. I insist upon my students' learning scales and arpeggios because they show up time and time again in their pieces, and if the student already understands the pattern of the major or minor scale, he or she does not have to relearn the finger pattern every single time it appears.
As for verbal ability, I suspect that might have something to do with the one-on-one instruction; I try to minimize talking in my teaching (and don't usually succeed), but my earliest lessons often involve learning the names of the parts of the violin and the bow, so the kids are getting new words that they would not otherwise need to know. Or old words in new ways; many small children find it extremely funny that part of the bow is called the "frog", even though it looks nothing like a green amphibian.
(Many small children then think the pegs are the pigs and the scroll is the squirrel, but that's a different lesson.)
Or maybe children whose parents enroll them in music lessons also have parents who place an emphasis on verbal ability in general. Correlation does not equal causation; all of these skill areas may be enhanced by the values of parents who also value music.
I do know that all three National Merit Scholars from my high school class were musicians. Two were string players, and one played trombone. And I'm quite sure that learning an instrument or proper use of the voice makes the brain and body work more efficiently together.
In short, enroll the whole family in music lessons!