Harriet the hairstylist can be found at Harv's Island Market. She can give you a new, random hairstyle, unlocking it as an option when using mirrors and vanities.
How to Unlock Harriet's Hairstyling Service
After you update the game to version 2.0, go to the airport on your island, tell Orville that you want to fly, and ask to go to Harv's Island. Go to the Harv's Island Market area behind Harv's house. If you haven't unlocked any of the shops yet, you will need to donate 100000 bells to one of the Lloids in the area. When the shop opens the next day, Harriet's hairstyling service will become available.
How to Get a New Hairstyle with Harriet
Talk to Harriet while the shops are open, and you can choose to get a new hairstyle. You will automatically sit in the chair and she will give you a random, Harriet-exclusive hairstyle. This will also unlock the hairstyle for use in mirrors and vanities, so you can switch to it any time. If you don't like the hairstyle, you can say "oh no" and you will go back to your previous hairstyle, but the new hairstyle will still be unlocked. You can also go into Harv's house and put a vanity or mirror in the room, and use it to change hairstyles.
There are seven new hairstyles that you can get from Harriet. After you obtain them all, Harriet will no longer stand by a stool near the bonfire at Harv's Island Market.