How did Mastercard get its name?

Mastercard was not named by the company that carries the name today. It was named by a bank in Kentucky.

In the mid-1960s several banks and regional bankcard groups joined forces to compete with Bank of America's BankAmericard, the card that preceded today's Visa. Wikipedia and Mastercard's own brand history both date the group, the Interbank Card Association, to 1966, though GoBankingRates says the card was created in 1967, and Mastercard's history says the Buffalo federation formed then. Early Interbank cards carried a small lowercase "i" in a circle.

The name Master Charge came next. Wikipedia says First National Bank of Louisville, Kentucky, coined it, and that the brand combined that name with the two overlapping yellow and orange circles of the Western States Bankcard Association. Wikipedia says the launch was in 1969. Mastercard's brand history says the name and graphics were adopted in 1968, with the name between two interlocking red and ocher circles and the "i" kept in the corner. That is a real conflict between sources, so this page gives both dates.

The full name was "Master Charge: The Interbank Card." In 1979 it became MasterCard, and Mastercard's brand history says the "i" symbol was retired in 1980. None of the sources read says why the name changed.

The circles proved more durable than the words. Wikipedia reports that MasterCard introduced a corporate logo with a third circle in 2006, changed its service name to a lowercase "mastercard" in 2016, and in January 2019 removed the name from the logo, leaving only the two overlapping discs. Mastercard's brand history likewise says the name was dropped from the mark for many uses, leaving the circles as a standalone symbol.

So the name is a loan from Louisville, the logo a loan from another bank association, and the one thing the company built itself is the network.