The 23-year-old Noah, whose mother Barbara Becker is the daughter of an African-American man and a German woman, took the decision in consultation with his father.
“I have been retained to quickly take the necessary steps under criminal and civil law against MP Jens Maier on the basis of this clearly racist tweet,” Becker family lawyer Christian-Oliver Moser told Bild.
Maier, a former judge who was one of nearly 100 members of the Alternative for Germany (AfD) party elected to parliament in the September election, had attacked Noah Becker over an interview in which he complained about being seen as the “eternal son” of his famous father.
“It seems the little half-negro simply got too little attention – that's the only explanation for his behaviour,” said the tweet posted from Maier's account on Tuesday.
It has since been deleted and Maier told Bild that not he but one of his staff members had written it.
It was the second time in a week that AfD deputies stirred outrage on social media.
German police on Tuesday filed a complaint against Beatrix von Storch, deputy leader of the party's parliamentary group, over a tweet on New Year's Eve which they say violated laws against incitement to hate.
Von Storch had criticized Cologne police for sending a New Year's greeting in Arabic on Twitter.
“What the hell is going on with this country? Why is an official police site… tweeting in Arabic?” she wrote. “Did you mean to placate the barbaric, Muslim, gang-raping hordes of men?”
The legal action came as an anti-online hate speech law went into effect on January 1st in Germany.
Social media companies that fail to remove illegal inflammatory comments could face up to €50 million ($60 million) in fines.
The AfD rode a wave of discontent against a mass influx of asylum seekers to Germany since 2015 to make the strongest showing for a far-right party in a national election in the post-war era.