That goal saw Kane equal Sheringham's tally and the England striker broke it 12 minutes later when he prodded the ball past Jordan Pickford from close range.
The former Bayer Leverkusen man said he wants to show better performance for his London club.
Inexplicably, Son is one of the most under-rated stars in English football and even at Spurs, he is not one of the club's A-listers, lagging behind Kane, Dele Alli, Christian Eriksen and keeper Hugo Lloris.
Son Heung-min opened the scoring for Tottenham at Wembley with his fifth consecutive home league goal - the first Spurs player to do so since Jermain Defoe back in 2004 - before Kane doubled their lead moments after the interval.
Alexis Sanchez doesn't want to play boring football at Manchester United
At a general level, what I can say is that myself and the Manchester United board, we do not believe a lot in the January market. Everyone is onto what you are trying to do in the market, and buying the best players costs an very bad lot of money.
This season has already seen Kane break Alan Shearer's record for most Premier League goals in a calendar year with 39, while also finishing 2017 with more goals than Lionel Messi and Cristiano Ronaldo.
Spurs continued their fine recent form as they earned a precious three points that puts them joint fourth in the Premier League table with Liverpool.
But the Toffees were cut to ribbons four minutes later as Aurier, Son and Alli all combined with one-touch passing to set up Eriksen, who drilled a first-time shot past Pickford from Alli's deft flick. I didn't expect to see it.
Everton had the ball in the net eight minutes later, but it was ruled out for offside.
In this photo taken by the EPA, Tottenham Hotspur's Son Heung-min (L) celebrates after scoring a goal against Everton during the English Premier League match between Tottenham Hotspur and Everton at Wembley Stadium in London on January 13, 2018.
YouTube Exec Addresses Logan Paul Suspension:"He Has Made Missteps"
In an open letter earlier this week, YouTube said "suicide is not a joke, nor should it ever be a driving force for views". The news comes as YouTube released a statement confirming that Paul would be punished for his misdeeds.
The South Korea forward has been involved in more goals at Wembley this season than any other Tottenham player, scoring seven and assisting four.
"It was a shock second-half performance".
62' - Dominic Calvert-Lewin replaces Cenk Tosun for Everton.
- This was Tottenham's biggest margin of victory in a game against Everton since a 10-4 win in October 1958. I saw some good signs of the players last week against Liverpool and I saw some of the worst signs in the second half today which is massively worrying.
Allardyce's side are now winless in their last five Premier League games and have lost their last three.
University of Rochester professor at center of sexual misconduct complaint apologizes
The complainants, at a news conference, called the report incomplete and said their legal fight would continue. The complainants said they took their action in support of students and were not to blame for the damage.