"They should give over the account details. Lucy needs justice. It's so easy for them to do."
MPs also voiced outrage, saying the delays in getting crucial information were ":deeply disturbing".
It has been almost six weeks since Lucy was stabbed to death and dumped in woods near a sports centre in Southampton.
Nicholson, 24, was swiftly arrested on suspicion of murder and of engaging in sexual activity with a child.
The suspect refused to give detectives the passwords to his Facebook accounts and was charged under the Regulation of Investigatory Powers Act with failing to co-operate with police. Nicholson, who was a lodger in Lucy's home, pleaded guilty.
Hampshire Police asked Facebook for access to his accounts in the early stages of the investigation.
The force was told it had to go through a lengthy bureaucratic process in the American courts and is still waiting for access.
Detectives believe the accounts could hold crucial information.
Miss White added of the police request: 'It would certainly give police an idea of what was being said between Lucy and Stephen.
"I will only rest when I have justice for Lucy, but until then I can't."
Miss White also criticised the social media company for turning her daughter's private Facebook account into a public memorial page.
"It's public so people can write whatever they like on there," she said. "They could at least have the decency to stop it from being public – I've messaged them about it and they still haven't changed it back."
This is the first time that Miss White has spoken publicly about her concerns over Facebook.
Last night senior MPs called for Facebook to do the 'right thing' to help police solve this "shocking case".
Yvette Cooper, chairman of the home affairs select committee, said: "This was an appalling murder, and Lucy McHugh's family need justice. For there to be such long delays and cumbersome international processes for getting crucial information in such a serious case is deeply disturbing.
"It shouldn't be beyond Facebook and the US and UK governments to come up with fast legal procedure for the police to get answers when it comes to investigating a child's murder."
Stephen Doughty, who also sits on the committee, said: "I would expect companies like Facebook to fully co-operate with law enforcement agencies, particularly in such a shocking case as this."
Earlier this year Lucy revealed on Facebook that she was "in a relationship" but did not say with whom. Nicholson, a care worker, told police he was in touch with Lucy via Facebook on the day she vanished and the night before – but would not let them see the messages.
Police were granted a court order requiring him to reveal the password, but he refused twice. The father of one claimed police would find information relating to cannabis on his Facebook timeline.
Last night police issued a fresh appeal for information. A spokesman said: "Sadly Lucy won't be returning to school tomorrow but her friends will be. One or more of them may have key information about what her plans were."
Last night Facebook refused to say if it had handed over the password. A spokesman said: "We are co-operating with Hampshire Police."
