She admitted that forcing users to post under their true identity would not stop online abuse but added: "It would certainly help a little bit."
"I think they should do more – they do have a responsibility for online," she told The Guardian.
Ms Rayner also claimed Facebook had told her fellow MP Stephen Doughty that "they had a higher bar for politicians" when he complained about the abuse and threats he had received.
“It is almost as if politicians should accept a certain amount of bullying and abuse online," she said.
“I try my best to stay true to who I am,” Ms Rayner said. “I don’t pronounce all of my words how they do on the BBC at times and that is OK, because I sound like the people I grew up with.”
Asked if she viewed herself as leadership material, Ms Rayner replied: "I think anybody can achieve if given the opportunity.”
Ms Rayner has also been mentioned as a candidate for a possible new post of co-deputy leader alongside Tom Watson.
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Shadow health secretary Jonathan Ashworth said his colleague was a "very good contender", adding: "Angela Rayner is great, she's a good friend of mine, a working-class woman, a trade unionist and former care worker."
Ms Rayner made headlines earlier this year after describing Labour's plans to increase public spending as a "shit or bust strategy".
Additional reporting by Press Association
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