Button, 36, is not retiring but said he will be taking a "break" next year.
McLaren followed up the 2009 world champion's announcement by saying they would employ a three-driver strategy next season, with Button signing a two-year contract extension in an "innovative role".
"To be clear, I'm very definitely not retiring. I'm contracted for both 2017 and 2018," said Button.
"I intend to work hard on car-development and I'm sure I'll get behind the wheel of the new car at some point.
"I'm delighted that I'll be staying on as a key member of the McLaren-Honda team."
He added: "Specifically, I'm looking forward to becoming even more deeply involved in the team's efforts to bring about the success we've all been striving to deliver.
"I love McLaren-Honda. I firmly believe it's made up of the best bunch of people I've ever worked with and I have no intention of driving for another team."
Button had been heavily linked with a return to Williams, where he started his career in 2000, following Brazilian Felipe Massa's announcement on Thursday that he plans to retire at the end of this year.
He is the third most experienced driver in F1 history with 297 Grand Prix starts -- only Brazilian Rubens Barrichello and German Michael Schumacher have made more.
The Briton has won 14 GP races.
'Brilliance and potential'
Vandoorne, 24, will partner two-time world champion Fernando Alonso on race days next season.
"I've signed loads of autographs in my career, but when I inked my signature on my McLaren-Honda contract I had an exceptionally special feeling," said the young Belgian.
"It's an enormous honour to become a McLaren-Honda race driver."
He has already made an impressive F1 debut as a substitute for an injured Alonso at the Bahrain Grand Prix, was the GP2 champion last year and has served as McLaren's reserve this season.
"McLaren-Honda's race driver line-up next season will be Fernando and Stoffel -- a perfectly balanced mix of proven brilliance and immense potential," said the team's CEO Ron Dennis.
About Button, he said: "Having extended his contract to include 2017 and 2018, he'll continue to be a senior, influential and committed member of the team, and will remain centrally involved in the development of our cars. He'll also be available to race for us if circumstances require it."
Dennis added: "I'm absolutely certain that the depth of his experience and the currency of his expertise will give us an advantage over our opposition next season."
Three-time world champion Lewis Hamilton led the praise for his former team-mate, having spent three seasons alongside Button while at McLaren earlier in his career.
"Jenson's been an extraordinary British champion," said Hamilton.
"I remember watching him when he first got to Formula One, on a weekend with my dad, eating bacon sarnies, watching Jenson doing what I was dreaming to do.
"He's done 17 years now, a long, long stint in this sport, but what he has given to it is fantastic."