The founding fathers didn't come up with it. The name "America" comes from the Italian explorer and cartographer Amerigo Vespucci, who would eventually become the Chief Navigator of Spain. He was the first to demonstrate that Columbus had not landed on East Asia, but an entirely new continent.

The German cartographer Martin Waldseemüller was in 1507 the first to use the name "America" to refer to the new world, deriving it as a feminine form of Vespucci's Latin name, Americus. It was already well established 269 years later when the US declared independence.

OP, can you clarify, are you asking (a) where the word "America" comes from (that is used not only for the country USA but also the continents North America and South America), or (b) why the "United States of America" was chosen as the name of the country? Thanks!