How do I make this pattern in Illustrator?

FINALLY !

An example to use that good-old and strange Twirl tool in the Reshape & Distort section !

Adobe Help: How to scale, shear, and distort objects in Illustrator

For the example below, I used the Rotation tool to copy a line with a varying line width profile 17 times at 20 degrees (I intentionally clicked the registration point point a bit off-centre, and used the option key to invoke the dialogue, and hit Cmd-D for 17 times...)

BTW: did you spot the Varying Line Width tool at the top, in case you need oddly varying line widths ?

Anyway, no matter how you create/repeat/duplicate the line's shape and transformation, this should be the result to continue with:

Then I took the Twirl tool (change its effective size by double clicking the tool first) and just played with it.

Using it off-centre as well created this very nice irregular effect:

Have fun !