The 23-year-old, who is not a member of police staff, was held on Wednesday night and has been released on bail. This follows the arrest of a police officer on Saturday.
Mr Mitchell quit as chief whip after it was alleged he called Downing Street police "plebs", which he denies.
CCTV footage has cast doubt on the original police accounts of the row.
The man was arrested in north London at around 20:00 GMT on Wednesday "on suspicion of intentionally encouraging or assisting an offence of misconduct in a public office on or around last Friday". His home was also searched.
BBC home affairs correspondent Danny Shaw said this had been "about the time police received fresh information about it [Plebgate]".
Prime Minister David Cameron, speaking during a visit to Afghanistan, said a "full-scale police investigation", supervised by the Independent Police Complaints Commission, was needed.
"I think that's very important. Let's get to the truth," he said.
By Gary O'DonoghuePolitical correspondent, BBC News
I understand that Andrew Mitchell has met Pat Gallan - a Deputy Assistant Commissioner, who is in charge of the Metropolitan Police's Directorate of Professional Standards, and the officer running Operation Alice, as it's been named.
I also understand that Mr Mitchell regards Ms Gallon as "extremely good" though he no longer has confidence in the Met commissioner himself.
That is because Bernard Hogan-Howe earlier this week publicly backed the two original officers who were present during the altercation in Downing Street.
There has also been what is described as an "unsatisfactory" exchange of letters between Mr Mitchell and the commissioner, which has further eroded his confidence.
The Met Police said they had no comment to make at this stage.
The offence allegedly took place a day before a Diplomatic Protection Squad officer, aged 52, was arrested on Saturday on suspicion of misconduct in public office.
But the Met confirmed the latest arrest was not over allegations of "conspiracy" to give false information about the argument between Mr Mitchell and police in September.
Thirty officers are working on the investigation - known as Operation Alice - which is looking into claims that someone on the force gave false evidence over the incident.
Paul McKeever, chairman of the Police Federation of England and Wales - which represents rank and file officers - said the organisation had taken "a very clear line" nationally not to call for Mr Mitchell's resignation, although he could not account for all 43 local federations.
Asked if he felt local federation members had been unfair to wear t-shirts reading "PC Pleb" and calling for the politician to be sacked, Mr McKeever told the BBC: "We have to wait for this investigation to take place to find out exactly what has happened - and we are supporting that investigation absolutely."
If Mr Mitchell had been done "a calumny" he would be "the first in the queue to apologise," he promised.
Mr McKeever added: "I understand those who say the federation stoked up some of the media attention in relation to Andrew Mitchell. I think we can all say we could have done things better."
But former Conservative home secretary Kenneth Baker said the federation was in "real trouble".
He told BBC2's Daily Politics programme it had "decided to become... an extremely aggressive lobbying body, determined to get a minister... because they were very opposed to what the home secretary is doing looking into their pay, pensions and early retirement."
Earlier this week Channel 4 News alleged that an officer had sent an email purporting to be from a member of the public who had witnessed the row.
The email was sent to Deputy Chief Whip John Randall, who passed it on to No 10. It contained claims that Mr Mitchell "shouted obscenities" and used "gutter language", adding that people watching were "appalled", with some possibly "inadvertently" filming what was going on.
But CCTV coverage suggests there were few, if any, members of the public within earshot of Mr Mitchell, the Conservative MP for Sutton Coldfield.
A senior Downing Street source told the BBC that Mr Mitchell was in a "much stronger position" following the latest developments in the plebgate story.
His friends have urged Mr Cameron to restore him to office, with some Tory MPs complaining that the prime minister should have moved more quickly to try to exonerate his former cabinet colleague.
Mr Cameron told the Commons on Wednesday: "A police officer posing as a member of the public and sending an email potentially to blacken the name of a cabinet minister is a very serious issue and does need to be seriously investigated."
And the Met says it is taking the issue "extremely seriously".
Mr Mitchell has admitted swearing at officers but denied calling them "plebs". He resigned from the government in October, following several weeks of criticism in the media.
A second man has been arrested in connection with the Andrew Mitchell "plebgate" investigation.