You ask about the city, but you might also be interested in another name the Romans called themselves: the Quirites.

The name itself is quite curious and has been fiercely debated over the decades. It might come from quiris, meaning spear. Some have said it came from the Sabines, as they had a city called Cures and came to be associated with the Quirinal Hill.

Romulus was worshipped as the god Quirinius, and we also have the Quirinalia festival.

If you are asking about the state, the most common and stable term is the name of the city, Roma. It appears on early Roman coins as the official name of the state. Sometimes we find Romano(rum) "of the Romans" - - this being the common substitution of the state name with the ethnonym.

From the I century BC we find, in coins as well as in inscriptions, SPQR, senatus populusque Romanus, i.e. "the Senate and People of Rome". The origin of this phrase is debated (is it "the union of the rulers and the common people" or "the rulers and the warriors" or "magistrates and people"?) but its use in ancient documents and texts is clear "the Government".

res publica and imperium Romanum did not, in ancient times, involve different kinds of government as today. Res publica is simply "public affairs", i.e. the State, and not necessarily a Republican, or Roman one. Imperium Romanum is the Roman power, both as the government and as the area of jurisdiction.

In late Antiquity a new name for the Roman Empire appears: Romania, "the land of the Romans"; the name will be kept by the Byzantines (Ῥωμανία), and is the origin of the name of modern Romania or the Italian region Romagna (which was an island of Byzantine rule surrounded by Germanic rule).

The Latin name is "Roma." "Rome" is an Anglicization. The name in tradition (Livy) is that the name derived from the founder Romulus. I don't buy that for a second, and I don't think there's a good answer any more than why Paris is Paris or Berlin is Berlin.

There is a tradition that says Rome's real name is Amor - Roma spelt backwards - which means love. I don't think anyone gives that any real credence anymore.