The jet was shot down by a Russian-made missile over eastern Ukraine in 2014, killing all 298 on board.
The West and Ukraine say Russian-backed rebels were responsible but Russia accuses Ukrainian forces.
The families' claim is based on the violation of a passenger's right to life, News.com.au reported.
The claim is for 10 million Australian dollars ($7.2m; Â£4.9m) for each victim, and the lawsuit names both the Russian state and its president as respondents.
Jerry Skinner, a US-based aviation lawyer leading the case, told News.com.au it was difficult for the families to live with, knowing it was "a crime".
"The Russians don't have any facts for blaming Ukraine, We have facts, photographs, memorandums, tonnes of stuff."
Mr Skinner said they were waiting to hear from the ECHR whether the case had been accepted.
The Kremlin said it was unaware of the claim, the Interfax news agency reported, but a senator with Mr Putin's party is quoted in state media as saying it was "legally nonsensical and has no chance".
There are 33 next-of-kin named in the application, the Sydney Morning Herald reported - eight from Australia, one from New Zealand with the rest from Malaysia.
Sydney-based law firm LHD Lawyers is filing the case on behalf of their families.
Flight MH17 crashed at the height of the conflict between Ukrainian government troops and pro-Russian separatists.
A Dutch report last year concluded it was downed by a Russian-made Buk missile, but did not say who fired it.
Most of the victims were Dutch and a separate criminal investigation is still under way.
Families of victims of downed Malaysia Airlines flight MH17 are suing Russia and its President Vladimir Putin in the European Court of Human Rights.