Materials:

Worsted weight yarn in the color of your choice
Crochet hook suitable for the yarn weight
Yarn needle
Scissors
Pattern:

Note: This pattern is for a newborn-sized bonnet, but you can adjust the size by using different yarn weights and hook sizes.

Begin by making a slip knot on your crochet hook.
Chain 4 and join with a slip stitch to form a ring.
Round 1: Chain 3 (counts as the first double crochet), make 9 double crochets into the ring. Join with a slip stitch to the top of the beginning chain 3. (10 stitches)
Round 2: Chain 3, make 2 double crochets in the same stitch as the chain. Make 2 double crochets in each stitch around. Join with a slip stitch to the top of the beginning chain 3. (20 stitches)
Round 3: Chain 3, make 2 double crochets in the same stitch as the chain. Make 1 double crochet in the next stitch, then make 2 double crochets in the following stitch. Repeat from * to * around. Join with a slip stitch to the top of the beginning chain 3. (30 stitches)
Round 4: Chain 3, make 2 double crochets in the same stitch as the chain. Make 1 double crochet in each of the next 2 stitches, then make 2 double crochets in the following stitch. Repeat from * to * around. Join with a slip stitch to the top of the beginning chain 3. (40 stitches)
At this point, you have completed the crown of the bonnet. The next steps will involve creating the bonnet’s brim.

Round 5: Chain 1, single crochet in the same stitch as the chain. Make 1 single crochet in each stitch around. Join with a slip stitch to the first single crochet. (40 stitches)
Rounds 6-9: Repeat Round 5.
You can adjust the number of rounds for the brim to achieve your desired length.

Fasten off and weave in any loose ends using a yarn needle.
To make the ties (optional):

Attach the yarn to one side of the bonnet.
Chain a desired length for the tie.
Fasten off and repeat on the other side of the bonnet.