My WAG is that “stocking cap” derives from the way they’re knitted or crocheted, in the same manner as thick winter stockings (and maybe of the same yarn?). But I don’t know enough about sewing to say if they are indeed made the same way. I have no doubt whatsoever that some Doper does know, and will pop in to tell me if I’m off base.
See option 3.
I’d never heard beanie for this object until two weeks ago on a trip to Colorado. A couple of members of my Texas family were calling them beanies, and I objected—until I saw them labeled that way in a souvenir shop.
I always thought beanies were hats with propellers, as in Beanie from The Beanie and Cecil Show.
I’d heard watch cap, but think I’ve always called it a knit(ted) cap.
Jughead wore a beanie sans propeller. And I’ve often read about college students (freshmen?) in the early 20th century wearing beanies.
Definitely not a stocking hat. When I was young (60s), folk used to wear long pointed stocking caps with tassels on the end. Definitely not the head hugging watch/knit cap.
My backpack buddies and I all call them Watch Caps.
If I’m thinking of the sort of hat the OP is talking about, then I agree with this. We didn’t get snow in my city, but you could travel a couple of hours and and be in the mountains. Hence, the only time I recall anyone wearing these is when skiing.
Knitted hat (for non poms) and pom pom (for hats with pom poms).
I have heard ‘beanie’ for the non pom pom version, but feel it’s a US import.
I’m in the UK.
Growing up in Western New York, we called them “winter hats.” Now I’m inclined to call them “knit caps,” but I’m not sure where I picked up that term.
knit hat/cap
The store I cashier at refers to them as “winter hats.”
Watch caps.
We’d say a Mike Monkee hat!!!
Forgot to leave the rest of my response…
I have NEVER heard of any kind of hat being called a toboggan - that’s a sled.
When I was a kid the kick was those long winter hats with tassels on the end - those were called tassel hats or stocking hats.
The standard knit hat with the rolled up cuff - winter hat or maybe knit hat.
To me, a beanie is a felt hat - like the Brownies (young Girl Scouts used to wear)
A toque - that’s Canadian
Watch cap - the warm hats sailors in the Navy wear
I’m in Minnesota
Watch cap. Or beanie. When we ordered team sweats this year, that’s what the supplier called them.
That’s exactly what comes to mind when I picture a “beanie”; specifically, Calvin flying around in his propeller beanie.
I used to call them “toboggans” during my childhood. Which was spent in my native northern Ohio.
Toboggan. I’m from West Virginia. I saw a few sites that say its southern, but then I saw a map of toboggan hat and it looks like it might be Central and Southern Appalachian. My mom is from PA and uses toboggan as a hat.
I would say though that we recognize both uses, but the hat is more common simply because toboggan style sleds aren’t typically used here. We usually use molded plastic sleds or innertubes or sometimes plastic bags or tarps. Very few runner sleds or toboggans.
Knit cap or ski cap.
But, a knit cap doesn’t typically have the tassel ball. I would not wear a cap like that. 
I have seen them in stores, but tassel balls and pom-poms seemed too dorky, even when I was 12.
Just my own personal style choice. YMMV
Funny, cause as a kid, the first such cap I remember wearing was my dad’s old navy issue cap from when he was a Seabee back in the 40s! 
Lost it somewhere, and have never since seen a hat as tightly knitted as that one.
I have never heard anyone use the word “toque” in my life around here. I have a feeling I would get weird looks or possibly a punch in the nose for using such “fancy words” around here. I would just call it a winter hat with a pom.
I grew up in Maryland calling it a watch cap or a wool hat. When I moved to Ohio, I found that the locals call it a toboggan.