The Scotland flanker emerged the winner following a public ballot.
Watson had an outstanding championship as Gregor Townsend’s side beat England at Twickenham for the first time in 38 years and France in Paris for the first time in 22 years.
The Scots also enjoyed a record win over Italy but lost home matches to Wales and Ireland.
Watson missed just 14 minutes over the course of the five matches. He scored one try and provided one try assist, carried 67 times for 321 metres, and successfully made every single tackle of his 55 in defence.
He was on a short-list of six for the award alongside Wales pair Taulupe Faletau and Louis Rees-Zammit, Ireland’s Robbie Henshaw and Tadhg Bernie, and last year’s winner, France scrum-half Antoine Dupont.
Watson, 29, is the second Scot to claim the award since its inception in 2004, joining Stuart Hogg, the winner in 2016 and 2017. The Edinburgh player won 35 per cent of over 125,000 votes cast.
Cockerill paid tribute to the openside and tipped him for Lions honours in South Africa this summer.
“He's been the stand-out for Scotland,” said the Edinburgh coach. “He’s been consistently good. He hasn’t just played one game then had an average game the next. He’s probably been Scotland’s best player the whole way through.
“We’re delighted for him. We’ve got the player of the tournament for the Six Nations playing for Edinburgh Rugby which is a credit to him and to the club too.
“Hopefully that will put him in a really strong position for the Lions tour coming up in the summer.
“I think he’s put himself in a very strong position to be part of that selection conversation.
“He’s probably in one of the strongest positions that the Lions can pick from so he cannot have done any more to put himself in the selection picture.
“No-one would be surprised if he gets selected for the Lions. However there are some other very good players from the other countries.
“But he could not have done any more that what he has and so let’s keep our fingers crossed that an opportunity comes along.”
Watson said: “I’m very surprised, obviously loads of top players were up for the award. I think when you do win something like that, it’s full credit to the team I’m in as well and to all of the guys who have won it previously.
“The Scotland team we’re all playing in at the moment is a great team and we really believe we can achieve things in the next few years.
"We think we’re heading in the right direction and full credit to the whole team, because without them, boys can’t win these personal accolades.”
Ben Morel, Six Nations Rugby CEO, said: “Our heartfelt congratulations go to Hamish on a fantastic campaign in what was a superb year for Scotland. He is a player who gives absolutely everything on the field and that has clearly resonated with fans.”