I saw this beanie and really loved it.
So I’m hoping to make a similar one myself – something that I never did before!
Should I find a pattern which resembles the shape, eg. Weekend Hat – PetiteKnit or How to Knit a Classic Beanie (Free Pattern and Video) - Sheep and Stitch and change the stitch pattern to waffle stitch for half of it?
Would appreciate any help! 
Good-looking hat. It looks like a variation on brioche stitch. If that’s true it may be a very stretchy stitch pattern. You’ll have to play around with gauge in order to get the size you want but starting with a basic beanie is a good way to begin.
I don’t do hats but if i did I think I’d start with a waffle knit pattern to get an idea of size and modify from there.
Maybe something like this
It’s a very nice looking hat.
Something like what @Creations posted could work, and then just knit plain stockinette after the waffle section for the top of the hat. Looks like you might want double the length of the ribbing so you have enough to turn up.
There are many variation of waffle or thermal stitch. The one in the hat you’d like to copy is a machine-knit stitch and you may not find a way to do it by hand so it looks exactly the same.
Hi there, love the waffle stitch on the Apricot beanie you posted. I googled how to do a waffle stitch and, as ColoCro mentioned, there are many versions. Tapped into salmonmac’s suggestion of brioche, something I’ve never done, brioche that is, not salmonmac’s suggestions (always take those on board) 
and found this.
Although it looks like 2 colours and different plies, I think the result is very similar. Worth a play I think…