To the ancient Greeks Asia would have generally meant what we consider to be Turkey and the Middle East. This would have included the famous city of Troy, from which our hero Paris hailed from. Basically she is offering to make him king of his whole continent in which he already lives.

The Judgement of Paris probably developed during what is known as the "Archaic Period" (or perhaps the earlier Geometric/Dark Ages, but the same applies) when Greece was relatively small, poor and very much marginal to the rest of the Eastern Mediterranean. The poetess Sappho, for example, frequently uses "Lydia", a region of Asia Minor, as a shorthand for luxury and wealth, and the phrase "richer than Croesus" refers to a ruler of Lydia who was famous in Greece for his wealth due to his patronage activities. During the later Classical period, this position was largely taken up by the Persians, so that when Socrates needs to describe how pleasant one version of death would be he says that even the Great King [ie, the ruler of Persia] would not know such pleasure. One of the very interesting aspects of pre-Hellenistic Greece is that it maintained a rather subordinate attitude towards Persia and was very aware of its own peripherality. On can presume (and this is largely confirmed by Tacitus) that the Germans told stories about their own great victories over the Romans in much the same way as the Greeks did with the Persians, but unfortunately these have not survived. Greece, and Israel to an extent, is thus one of the few examples we have of literature produced from outside of a "core" region from the ancient world.

Which is all a way of saying that Paris was offered Asia because to a Greek at the time that might as well be being made king of the world.

Tiako's and nilhaus' answers are fine, but just for the record, the offerings that you mention in your question are a synthesis of several different accounts. In the earliest source to report on exactly what each goddess offered, Euripides' Trojan Women 925ff., Athena offers him kingship over Phrygia and the destruction of Greece; Hera promises him dominion over Asia and Europe, i.e. both Asia Minor and Greece. This isn't quite all the lands occupied by Greeks: it doesn't include Italy/Sicily or Olbia (in what is now the Ukraine), so their absence from Hera's promise may hint that this is actually an old legend pre-dating Greek colonisation in those places; alternatively, it may simply be Euripides imagining an ancient time when there were no Greeks in those places.

The promises vary from source to source. In the scholia to the above passage, for example, Athena promises him kingship over Greece; in pseudo-Hyginus Fabulae 92, Hera promises that he will rule over all lands, while Athena promises to make him the strongest of mortals and possessing all skills. "Wisdom" is a relatively recent simplification.