For us grown-ups, the benefits of keeping the mind active are well documented.
Logic puzzles like Sudoku are very good at keeping us thinking logically, and indeed, a daily dose of puzzle-solving can help us in many aspects of our lives.
Kids haven't fully developed their logical thinking skills yet, so their problem isn't keeping their minds sharp, but more to develop their logical thinking in the first place! This is one of the many reasons why Mathematics is taught at schools, they might think they are learning about fractions, decimals, algebra and all sorts of weird and wonderful things.
And they are, but they are also learning to think in a clear and logical manner! By working on Sudoku puzzles as well, they can further develop those logical thinking skills.
Standard 9*9 Sudoku grids will probably be too hard for them, at least initially.
But some websites publish smaller grids as well.
You can find 4*4 puzzles to start off with, and then move on to 6*6 puzzles.
And after that they can even try some easier 9*9 puzzles! The great thing about these smaller Sudoku puzzles is that the rules are exactly the same, they are just a smaller grid.
So, if you know how to work out the 9*9 puzzles, you already know how to work out the 4*4 puzzles and the 6*6 puzzles.
You just apply the same rules! I know young school kids (5-10) enjoy the 4*4 Sudoku puzzles.
It gives them a familiarity of working with numbers, and they are actually quite easy; as an adult you will be able to solve them very quickly.
Older kids (11+) can be working on the full-size 9*9 grids, but they might also be interested in KenKen.
This is a closely related puzzle, but introduces a few rules of Mathematics as well.
They tend to be smaller, but they are actually much harder, and consequently much more satisfying!