I followed Pfirsch's advice and tried the Saranac stitch.

This was supposed to be the gauge, but I thought it was ridiculous to crochet a square and then frog it or leave it aside. My daughter saw it and asked ifit was a purse for her...

I said yes

Follow up:

So here's my daughter's "Pacifier holder" or porte-tute ("tute" is the word she's been using instead of "tchuque" since she was able to utter something,I suppose people from Paris say "tétine", but we, Midi people say otherwise).

we go for the gauge:

Gauge : 10 X 10 cm = 27 stitches X 9,5 rows

Yarn : Bouton d'or "flash" cristal (light blue) and bourgeon (light green), 88% viscose, 12% polyamide, 50gr/125m, 1 3/4 oz/137 yards.

Crochet hook : n°3 (USA: D, UK: 10)

I used two different colors just to see what it would look like, and I like the effect. I change colors very much in the way it's done for the interlocking filet crochet technic, i.e the first yarn waits with a stitch marker while I crochet the second one, and vice-versa... I think both right side and wrong side are pretty!

To make the strap I used this very simple twisting method, using four strands of yarn of each color. The button is a big glass bead, the button hole is just a chain reinforced with single crochet.

To make the purse, at the end of the gauge I increased slightly to be able to fold the top and create an opening/closing part, and I crocheted the same number of rows minus 2 (to have sides). then i sewed the bottom and one side, attached the bead, made the button hole, et voilà!

She's very happy with it, and so am I!

edited to add :

I was asked how to make the strap, so here's a how-to... tell me if I'm not clear :p

1. first you choose the length of your strap : you need at least four time more
2. cut the required length of yarn and fold it once
3. fold it gain and insert a pen in the fold while you hold the other end
4. if you're alone, tie the end somewhere high enough to be able to grab the pencil, let it hang and make it rotate (like the blades of a chopper), when the strand of twisted yarn begins to swirl upon itself, pull it holding the pencil and continue twisting
5. if you got help, have one person holding one end while you twist the other one, holding the pencil; when the strand of twisted yarn begins to swirl upon itself, pull it holding the pencil and continue twisting
6. when you have enough tension, grab the strand at the middle, pull the middle very tightly and fold the strand in two
7. when both ends are joined, release the middle
8. smoothe the strap running your fingers from end to end
9. you're done

Happy crocheting!