The complaint filed by Arizona resident Jacqueline McAferty in the Austin, Texas federal court on Tuesday said Musk and his America PAC organisation falsely induced voters to sign by claiming they would choose winners randomly, though they were predetermined.
She also said the defendants profited from the giveaway by driving traffic and attention to Musk's X social media platform, and by collecting personal information such as her name, address and phone number that they could sell.
A lawyer for Musk and lawyers for McAferty did not immediately respond to requests for comment on the complaint.
McAferty sued one day after a Philadelphia judge denied a request by that city's district attorney Larry Krasner to end the giveaway, which Krasner called an illegal lottery.
That ruling was largely symbolic because Musk has no plans to give out more money following the US presidential election.
The world's richest person opened the giveaway to voters in seven battleground states who signed a petition to support free speech and gun rights.
Tuesday's lawsuit seeks at least $US5 million in damages for everyone who signed.
Musk is a Texas resident and his electric car company Tesla is based in Austin.
He has supported Republican Donald Trump in the presidential race against Democratic Vice President Kamala Harris.
