From the Ground Up
You will need to have begun the Hylian Homeowner sidequest first, which you can learn about here.
While in the middle of working on improving your house, Bolson will stop offering upgrades and talk about how Hudson is getting ready to leave soon, so you should talk to him. Talk to Hudson and you will start this quest with him leaving for the Akkala region to start a new town.
The town in question is located in northern side of Lake Akkala atop a column of rocks and is connected to mainland by a long bridge. The easiest way to get there is to Paraglide from Dah Hesho Shrine, or if you haven't unlocked it yet, the site is directly east of Akkala Tower.
Once you get there, talk to Hudson who is working away at rocks. It seems like building a town from scratch is harder than it seems, but you are more than happy to help. This sidequest is long and alternates between giving Hudson bundles of wood and finding him people who can populate the town and help it grow.
First, Hudson will ask for 10 bundles of wood. This should be pretty easy. In fact, you can even just cut down the trees that fringe the Tarrey Town site.
Next, Hudson will ask for a Goron who can help move all the rocks out of the way, preparing the land for construction. There is a catch though, and this will apply to all NPCs that Hudson will ask you to find. Their name must end with "-son". Just like Bolson and Hudson.
With that in mind, go to the Southern Mine in the Eldin Region just south of Goron City. You will have to arrive at night when all the work is finished. Speak to Greyson who will ramble about how boring work at the mine is. Tell him about Tarrey Town and he will leave.
Return to Tarrey Town and Hudson will ask you for 20 bundles of wood. At this point to farm up wood fast, you can go to the forest near the Old Man's House on the Great Plateau or go to the Korok Forest, or any of the vast forests in the Faron region.
Deliver the wood and Hudson will ask for somebody who is good at mending and making clothes. More specifically, Hudson would like for you to find a Gerudo. Travel to Kara Kara Bazaar in the Wastelands region and speak with Rhondson who is sitting under a tent alone. She will talk about feeling unfulfilled and leave for Tarrey Town if you tell her about it.
Again, speak with Hudson. The amount of wood he needs has gone up again, to 30 bundles of wood. Get to farming!
Deliver the wood and Hudson will ask you to find him a Rito who can use their wings and business sense to bring more merchants and shops to Tarrey Town. Go to the Rito Village in the Tabantha region and speak with Fryson. He is sitting by himself on the lower end of the Rito Village and will talk about how he wants to grow businesses and start his own instead of manning his parents' shop. He will leave for Tarrey Town.
You should follow suit, because Hudson needs more wood as more people are beginning to move into Tarrey Town. This is the last time Hudson will ask you for wood, but it's the biggest order yet: 50 bundles of wood! Once you have enough, deliver them to him for a final request.
It seems that Hudson and Rhondson have found love. They will be getting married (awww) but they need an officiant. According to Hudson, finding the perfect, priestly officiant means finding a Zora. Of course, this Zora's name will have to end in "-son". Travel to the Zora's Domain and speak with Kapson, an older Zora in the room behind the throne room. He is more than happy to officiate the ceremony.
Speak with Hudson again and there is just one last thing: invite Bolson and Karson! They are still sitting by your house at Hateno Village, so head on back there and let them know that their friend is getting married. Once that's done, you can travel to Tarrey Town and speak with Hudson to begin the ceremony.
With Tarrey Town officially complete and Hudson and Rhondson officially married, speak with Hudson once more to receive your prize, three Diamonds (!!!) and all the shops Tarrey Town has to offer.