The 35-year-old, whose sole world title came in 2009, has a contract that runs until next year but he had been considering his position in the sport in recent weeks.
"Over the past month or so, I have done quite a lot of thinking and it is no secret that I was at one point in two minds about my future," the Englishman said in a statement.
McLaren, who lie ninth out of the 10 teams in the constructors' championship after a difficult season, could have terminated Button's contract after this season should they have wanted to do so.
Instead McLaren CEO Ron Dennis, who has held several talks with Button, appears to have succeeded in persuading the driver that the marque has a brighter future.
"The fact that our talks have led to today's announcement is very pleasing to both of us and will delight and motivate all at McLaren-Honda," said Dennis.
"There is a 'terminate after year one' option that McLaren could have triggered, but once it became clear from many conversations with Jenson that he remained as enthusiastic, committed and focused as ever, that option immediately became an irrelevance."
After a difficult year with Honda, Button and teammate Fernando Alonso -- a two-time world champion -- are languishing in 18th and 16th place respectively in the standings.
Honda is in its first season back in F1 since 2008 but has struggled for power and reliability this season.
After finishing 11th in the Japanese Grand Prix on Sunday, Alonso described the car's performance as "embarrassing" over the team radio, comparing it to a GP2 engine.
However, Button believes that Dennis, who has been with McLaren since 1980, has the ability to restore the brand to former glories.
Both would welcome a return to the days of a five-year partnership between McLaren-Honda which powered Ayrton Senna and Alain Prost to a clean sweep of world titles from 1988 to 1991.
"[Ron] and I have had some very good chats these past few weeks, and during those chats it has become to clear to me that Ron is both utterly determined and uniquely equipped to lead our team through its current difficulties to great successes in the future," said Button.
"That gives me great confidence, and it is for that reason that, together, he and I have decided to continue our partnership.
"As soon as I had made that decision, straight away I realized it was the correct one."
Dennis, meanwhile, is delighted to have secured the services of a driver whose first F1 race came back in 2000.
"Jenson is the most experienced driver currently racing in Formula One, and next season he is due to become only the third driver (after Rubens Barrichello and Michael Schumacher) in Formula One history to pass the milestone of 300 Grands Prix," said Dennis.
"Although that wealth of experience makes him a massively valuable asset to our team, hugely expert in all aspects of the 21st century Formula One driver's craft, he is also supremely fit and as super-fast as ever."
No mention was made of whether Alonso will continue with McLaren next season, with the Spaniard refusing to be drawn on his future after Sunday's race at Suzuka.
Both Alonso and Button are more used to fighting at the front of the grid than the rear, and Button will be desperate that McLaren-Honda can find the extra edge next season.
"Granted, this year has not been an easy one for us, but we know what we need to do to improve things," said Button, who started racing for McLaren in 2010.
"In collaboration with Honda, we will work extremely hard over the next weeks and months to make sure that 2016 will be a much better season than 2015."
The second most successful team in F1 history after Ferrari, McLaren have not won a grand prix since 2012 -- when Button won in Brazil.
He has scored just six points in 14 races this year, while suffering repeated penalties and retirements.