Wikipedia's Quasar includes the following in its introduction:
The term quasar originated as a contraction of quasi-stellar (star-like) radio source, because quasars were first identified during the 1950s as sources of radio-wave emission of unknown physical origin, and when identified in photographic images at visible wavelengths they resembled faint star-like points of light. High-resolution images of quasars, particularly from the Hubble Space Telescope, have demonstrated that quasars occur in the centers of galaxies, and that some host-galaxies are strongly interacting or merging galaxies.
So today instead of "quasi-stellar" we might simply say "unresolved".
The light is star-like because the light from galaxies has a strong stellar component. (this second sentence is argued against in comments)
Why are quasars so far away that they required high resolution techniques (i.e. space telescopes or adaptive optics) to resolve them? Is it just that they are fairly rare and a roll of the dice resulted in none close enough to be resolved in older large telescopes of the 1950's, or are there some cosmological effects in play as well?