pŏpīna is the one, borrowed from Oscan or Umbrian, and cognate with (native Latin) coquina. Indeed, a Packhum search gives no results for thermopolium and 54 results (59 matches) for popina - note that, still, thermopolium can be found e.g. in Plautus and Petronius.
Note that caupona and taberna were also common, but while L&S lists them as synonyms, they meant something slightly different. The former was more precisely a place where you could also sleep. The latter term, if not qualified with a following adjective (libraria, unguentaria, etc.), indicated a place where only or mostly wine was offered.
Here is an exhaustive excerpt from Gustav Hermansen's "Ostia: Aspects of Roman City Life" (he gives credit to Tönnes Kleberg in the first line):
The main result of his research is that there are six important names for establishments in this category: caupona, deversorium, hospitium, popina, stabulum, and taberna. These six words are distributed three ways:
The caupona, deversorium, hospitium, popina and stabulum denote what modern languages would call hotels: houses where travellers could put up for the night and at the same time could have their meals and could drink. A stabulum originally meant a stable, and in this connection it stands for a hotel where one could stop in with one's horse. As Latin literature shows, however, it did not matter whether one brought a horse or not!
A caupona was a hotel which catered to all of a traveller's needs, but eventually the value of the word was down-graded and acquired the connotation of a low-grade saloon, then was dropped from the language altogether.
A popina was a restaurant which could serve food and drinks to a customer but which did not offer any accomodation.
A taberna was comprehensive, meaning a shop in which all kinds of trades could be plied. To differentiate, the word was qualified with adjectives like (taberna) libraria, unguentaria, vinaria, or, [as seen above], cauponia. Eventually the meaning of a drinking place (tavern) prevailed when the word was without an adjective. A taberna would normally offer wine, rarely food.