I've seen several patterns on Pinterest for knitted and crocheted dress-up crowns. It's a great idea, but I want to share a secret that most of these crowns share, whether knitted or crocheted: They are almost all made from traditional edging patterns!
A lot of these crowns are made from edging patterns that are worked short-ways, creating a smooth edge on one one side and a peaked edge on the other. It's the sort of pattern that our grandmothers and great-grandmothers would have used to edge pillowcases or shelf edges. If you have a pattern booklet of edging patterns, you have an inexhaustible supply of crown patterns. Lace edging patterns for both knitting and crochet are also available for free all over the internet, if you are willing to sift through a Google Image search to find what you want. Knitting-and.com has lots of options. You can also improvise your own pattern.
Choose a smooth yarn, such as cotton, or crochet cotton in size 5, 8, or 10, and a small enough hook relative to your thread or yarn to make the fabric a little stiff. Most of the patterns I've seen online use yarns ranging from worsted weight wool or acrylic to sock yarn, but I think the crisp look of cotton and a fine gauge looks best. If you want something fancy, carry a metallic thread along with your yarn or choose a yarn that has a little sparkle to it. You can even thread some beads onto your thread to include along the smooth edge or at the points on the peaked edge (or both). If you want to get really fancy, you can choose a really simple pattern, but make it out of thin jewelry wire and beads.
Make the edging long enough to fit snugly around the recipient's head, and then sew the beginning to the end. That seam should be worn in the back, and, if desired, you can attach a piece of tulle or some ribbons there to cover it up. You can leave the tulle or ribbons long enough to trail down your little princess's back.
I made this one for my daughter using a fingering weight, cotton/acrylic blend and a size C (2.75 mm) hook. I used the pattern at this link (there are tons of crocheted edging patterns available for free at the Free Vintage Crochet website), but left off the heading row to keep the crown relatively elastic. Making it took an afternoon, and my daughter loves it.
For something fairy-like, find a mini-snowflake pattern (This link goes to a pdf of an old pattern booklet for crocheted miniature snowflakes. My mother used to have this booklet in hardcopy, and I've made several of the patterns in it). Make motifs in white, with iridescent or silver beads at the points. Join the motifs at the points in a long strip to fit around the little snow fairy's head. Alternatively, make several flowers and leaves, and arrange them into a daisy chain.
If you are concerned that a crown is a little too big, thread a piece of ribbon all the way around the crown, leaving long ends in the back. When the crown is put on, the ribbon can be tied to cinch the crown and adjust the fit.
If the edging you use sticks up far enough that it flops over, you can starch it, but know that the crown will not be elastic anymore if you do that.