Incomplete lists were delivered to all 155 stations and many of those trying to vote before going to work were unable to do so.
The council later sent out the correct versions and urged would-be voters to return.
It has promised a "thorough investigation".
Labour's Andrew Dismore has held the Barnet and Camden London Assembly seat with a 16,240 majority over the Conservative candidate.
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The council offered emergency proxy votes to residents who were turned away and unable to return.
They included the Chief Rabbi, Ephraim Mirvis.
Candidates and voters can challenge the result of the elections in London by lodging a petition with the Royal Courts of Justice.
Grounds for the appeal include claiming the successful candidate was not duly elected, or the election was invalidated by corrupt or illegal practices.
Sophie Walker, the Women's Equality Party (WEP) mayoral candidate, registered a complaint with the council as well as the London Assembly.
A WEP spokeswoman said: "We are waiting for more information before we decide what our next steps are."
The Labour leader on Conservative-controlled Barnet Council, Barry Rawlings, told the Guardian the first set of lists delivered to polling stations contained only the names of people who had registered since January.
"It would have been 10% the size of the proper list. They should have spotted it."
A Barnet Council spokesman said: "We cannot comment any further on the impact of the issues with the electoral registration lists yesterday. We will begin a thorough investigation into the cause of the problems once results are declared."
Barnet Council may face a legal challenge into the blunder that led to some voters being turned away from polling stations on Thursday.