Google free knitting patterns or free knitting patterns children and you'll see many sources of hundreds or thousands of free patterns. [I don't knit, but I crochet, and I could keep busy for several lifetimes on the great crochet patterns I've found.]

So my advice would be to find a free hat pattern that appeals to you. Make the hat, then crochet a rectangular scarf which uses stitch elements from the hat pattern. [Perhaps you'll see a pattern for a similar buttoned scarf which will give you the size to use. Or perhaps you and the intended recipient can experiment with different sizes. Or if the intended recipient isn't available to test sizes, perhaps you know another little girl who can step in as a model.]

Buy a big button that looks good with the yarn. Then on the last edge of the scarf, make a buttonhole (however knitters do that *lol*).

[Besides regular button sources, my WM has an area in the fabric/crafts section with kid-appeal buttons (star-shaped and I-don't-remember-what-all). These were sold in small and large packets.]

Good luck with the project!