This term has a long historical context, first being used in France in the 1800s. It seems to have switched sides many times, and been flexed and twisted to mean a lot of different things.
In contemporary Western politics, it mainly refers to the resurgence of conservative (yes, conservative) politics in the late 20th century. These politics involved policies like deregulation, economic austerity, privatization, etc. – anything that acknowledged and enabled the supremacy of the free market.
Ronald Reagan and Margaret Thatcher are the most representative neoliberals, and the ones with whom the label is most associated.
So, where did the name come from? If it’s associated with the word “liberal,” it’s odd that it refers to conservative politics.
This page has a lot of history of the term (although, with a conservative slant).
From my reading, I don’t see any absolute agreement on the etymology of the term. Does the “liberal” part refer to a political position, or the idea of “liberty”? Or does it mean a “new liberal,” meaning a liberal who has changed positions? Or does it mean “new liberty”?
(This is further complicated by the existence of the term “classic liberalism,” which is essentially the opposite of what we consider a “liberal” in contemporary politics.)
This page acknowledges the confusion by quoting an author:
[Neoliberal] is now widely acknowledged in the literature as a controversial, incoherent, and crisis-ridden term, even by many of its most influential deployers.