Finch joined Boro as an Under-14 and went on to make six senior appearances for the first team, the last of which came in the Carabao Cup defeat to Stoke City at the Riverside at the start of last season.
Former boss Michael Carrick rated Finch highly but a loan move to MK Dons last season didn't go to plan, and his contract was due to expire this summer.
Finch linked up with South Shields on loan in January and has starred for the National League North promotion hopefuls, scoring six goals, the best of which came last week with a stunner against Merthyr Town.
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He has now put pen to paper on a two-and-a-half year deal with Shields.
Finch said: “I’m absolutely delighted to get this done. I’ve really enjoyed my time here over the last few months, so making it permanent was an easy decision for me.
"From the moment I arrived, I felt at home. The lads, the staff and the fans have all been brilliant with me, and that made me want to stay. It’s a great club with real ambition, and I’m excited to be part of what we’re building here.
"I feel like I’ve developed a lot already during my time here, and I know there’s still more to come from me. I just want to keep working hard, contributing to the team, and hopefully we can achieve something special together.
"I can’t wait to get back out there and continue where we left off.”
Boss Watson said: “We’re delighted to bring Sonny into the football club on a permanent basis. He’s a player with real quality and potential, which he has demonstrated during his time here.
"This is an incredible signing for the football club and shows the ambition we have here. We’re all working as hard as we can to make South Shields FC the best football club it can be.
"Sonny wanting to sign here is a credit to the players and the culture they have created for players to really want to be part of what we’re building.”
Boro said in a statement: "Everyone at MFC wishes Sonny every success for the future."