Shirley Yamauchi said she hopes her lawsuit will help protect other passengers from this type of treatment.
"United said they would change and I want to see that happen. I don't want anymore passengers possibly in danger," she said.
Her attorney says he is ready for a trial.
"United deserves everything we can do to them. We'll let the people decide what to do to people that are this greedy and put lives potentially in danger," attorney Michael Green was quoted as saying by the Hawai News Now.
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As children over the age of two years are required to have their own seat, she purchased a ticket for her son, Taizo - paying almost USD 1,000 each ticket.
When she brought the problem up to flight attendants, Yamauchi said she was ignored.
The airline did not apologise or respond to her until five days later, saying agents inaccurately scanned Taizo's boarding pass, the report said.
"It could have been resolved at that point, yet not a single flight attendant asked why he was on my lap the whole duration of the flight," Yamauchi said.
"There was a representative waiting for us. We were escorted to their United lounge, but we explained we don't have a membership with them, yet they asked us to utilise their lounge. The three of us were moved to business class after getting our tickets. He offered to carry my bag for me down to baggage claim. We both received a lei. A flower lei," she said.
When asked for comment about the potential lawsuit, United said, "We reached out to Ms Yamauchi to apologise to her and her son, and we also refunded their tickets."
In April, a passenger named David Dao refused to give up his seat on a United flight and was dragged, battered and bloodied down the plane aisle by police.
Navang Oza, an Indian-origin man in the US, claimed that the United Airlines cancelled his flight reservation because he filmed an employee during an altercation.
A giant rabbit, who was expected to outgrow his father to become the world's biggest bunny, also died under the care of United Airlines earlier this year.
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