The New Deal surely did deliver a “new deal” to the American people. It fundamentally shifted the relationship between people and the government.
Before the New Deal, people did not look to the government for help. For example, in 1887, Texas farmers hit hard by a drought appealed for aid from the government. The Congress voted to appropriate money to help them, but President Grover Cleveland vetoed the bill saying
though the people support the Government, the Government should not support the people.
This was the general attitude of most Americans and most American political leaders up until the time of the Great Depression.
With the coming of the New Deal, all that changed. The government now proposed to help support the people. It proposed to ensure that they would not be harmed excessively by economic bad times. It proposed to insure the safety of their bank deposits. It proposed to make sure that they would have pensions in their old age. It even proposed to ensure that farmers would not produce too much and thereby drive down prices.
All of this constituted a radical change (a “new deal”) in the relationship between people and the government.