Patricia Devlin, an award winning crime reporter with the Sunday World in the North of Ireland, is taking the case to the Police Ombudsman.
Her lawyer has claimed there is “more than ample evidence” to arrest the suspect.
The reporter has the support of Amnesty International.
A PSNI spokesman said: “PSNI encourages anyone with a complaint regarding police actions to contact the Police Ombudsman.”
The mother-of-three received the threat a year ago in a direct message to her Facebook account, signed in the name of neo-Nazi group Combat 18.
Ms Devlin said she told the PSNI who she believed the individual responsible was.
The reporter states he was a convicted criminal who had previously been involved in violence.
“The police have had this individual’s name all this time, yet, a year on, no-one has been brought in for questioning, never mind arrested.
“Meanwhile, the police have given me a constantly changing and contradictory story as to why they have not acted,” she added.
The journalist said because of her job she had been dealing with threats for years, but “had enough” with the threat to rape her newborn son.
The message also identified the location where the sender believed Ms Devlin’s grandmother lived.
Ms Devlin’s lawyer, Kevin Winters of KRW Law, said she had no option but to lodge the complaint, claiming explanations for the police not making an arrest “border on the farcical”.
Patrick Corrigan, of Amnesty International, said: “The complete failure of the PSNI to properly investigate this case is totally unacceptable.
“Amnesty International has been watching with increasing concern the constant stream of threats being received by journalists in Northern Ireland, designed to shut down press scrutiny of criminal and paramilitary activity.”
The National Union of Journalists is also supporting Ms Devlin’s complaint.
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