Usually people who designate or "discover" a constellation name it and that name is passed down through history. As a result there are many different names for many different constellations. For example, the constellation we now refer to as Scorpius was referred to as "Ip" in Egyptian hieroglyphs from before 3000 B.C. Since there were so many different names for the same constellations, in 1930 the International Astronomical Union made an “official” map of the sky and designated 88 constellations and names. Most of the "standard" constellation names we use now are Latin, like Scorpius. Many of these Latin names originated from the Roman empire, although some are more modern. Some are named for figures in Greek mythology, like Orion. In short, there are a lot of different origins for different constellation names!

They were named by the Babylonians and the Greeks, based on mythology.

There are 88 constellations officially recognized by the International Astronomical Union (IAU). Of these, 48 (actually 50 - Argo was divided into 3) are "ancient" constellations attributed to Middle Eastern, Greek, and Roman cultures and recorded roughly 1900 years ago around year AD 150 by Claudius Ptolemy. There are an additional 38 "modern" constellations described by astronomers in the 1500s, 1600s, and 1700s after they began to study the sky of the southern hemisphere and the telescope was invented. In all cases the names are descriptive of the pattern formed by the stars. Constellations have both a Latin name and a translated name (for example, the constellation with the Latin name "Aries" is also known in English as "the Ram").

Unless the IAU changes how constellations are defined, there will not be any more official constellations which will need new names. To make the current official set the IAU divided the sky sphere surrounding Earth into 88 pieces, each piece being referred to as a constellation and enclosing one of the 88 star patterns. Since these pieces occupy the entire sky, and (official) constellations can't overlap, no more can be identified and named. This division was in part to aid in naming of stars that were being identified with new technology. Note that constellation (per the IAU) refers to the patch of sky surrounding the star patterns, and that all stars within that patch of sky are said to belong to that constellation even if they are not part of the figure from which the name is derived. A closely related term is "asterism". Asterisms are patterns or shapes that are widely recognized but are not part of the official constellations. One example is the Big Dipper, which is part of the constellation Ursa Major. (Info from the IAU and the Lunar and Planetary Institute. )

There are new unofficial constellations occasionally announced though. In 2018, NASA announced 21 "constellations" created by features in gamma-ray images taken by the Fermi telescope. These were identified and named by the scientists of the agency and include Einstein, a Saturn V rocket (used in the Apollo missions), and Mjolnir (the hammer of the god Thor of Norse mythology). "The new gamma-ray constellations and the official visible light constellations can be explored using NASA Fermi's space map."

Here's a good answer from the Lunar and Planetary Institute. I gave a talk at one of their conferences last year. The conference was about the first billion years on Earth.

How are constellations named? Most of the constellation names we know came from the ancient Middle Eastern, Greek, and Roman cultures. They identified clusters of stars as gods, goddesses, animals, and objects of their stories. Constellations.