I think that maybe you are supposed to really think about your business name. You are supposed to choose something that is easy to remember, easy to spell and meaningful.
I went with the fun factor! But you maybe wondering why Unicorns? I didn't really set out to be the unicorn lady it just sort of happened.
Once upon a time there was a girl who loved all things colorful and sparkly. She decide that she should really get in shape so she joined a Crossfit gym. (isn't that a beautiful beginning?) During that time I had a super cute unicorn shirt and a pair of deadlifting unicorn leggings and I became known as the unicorn girl. At that time (due to some injuries) I spent a lot of time rowing and the rower I usually used ended up decorated with sparkly unicorn stickers. We did secret Santa and I got unicorns. Unicorn shirts, unicorn socks and stuffed unicorns. It just became my thing. 
In my personal spinning I like bright colors and sparkle. So when it came time to give my fiber business a name, Unicorn Fluff just seemed like a fun fit! My friend Tori helped me with my branding and she drew all of the unicorns that pop up on my social media and signs. If you look close Sparkle the Unicorn has a handspun mane.
I do often get asked (and some of them are serious) where I get the unicorn fluff and so I will give you a little bonus lore..........On a full moon midsummer night the unicorns come out to play in their favorite woodland meadow. This meadow is a secret as it is next to my favorite huckleberry picking patch. After the unicorns finish their midnight frolic they lay down to rest and if you happen to bring their favorite snack-huckleberry bread-they will let you take a little bit of their magical fiber. 
Treleven thinks she is a unicorn.
And that is how Unicorn Fluff got it's name. :)
Flufffilly yours,
Shannon