The Italian Wikipedia page on Roman onomastics states, without references though,
Former auxiliary soldiers and other categories of people that earned the Roman citizenship, could and often would keep at least a part of their original name. A good many names (cognomina when assuming the tria nomina, but old personal slave names) were of Greek origins, while others came from the lands in the sphere of control of the Romans. Auxiliary soldiers that weren't Roman citizens, once having gained the right, often adopted the nomen of their Emperor, adding their original name as cognomen.
The corresponding English page also states:
Customarily a newly enfranchised citizen would adopt the praenomen and nomen of his patron; that is, the person who had adopted or manumitted him, or otherwise procured his citizenship. But many such individuals retained a portion of their original names, usually in the form of cognomina. This was especially true for citizens of Greek origin.
The Constitutio Antoniniana promulgated by Caracalla in AD 212 was perhaps the most far-reaching of many imperial decrees enfranchising large numbers of non-citizens living throughout the empire. It extended citizenship to all free inhabitants of the empire, all of whom thus received the name Marcus Aurelius, after the emperor's praenomen and nomen. The result was that vast numbers of individuals who had never possessed praenomina or nomina formally shared the same names. In turn, many of the "new Romans" promptly discarded their praenomina, and ignored their nomina except when required by formality. As a result, the cognomina adopted by these citizens, often including their original non-Latin names, became the most important part of their nomenclature.
The sources for this excerpt are the 1970 edition of the Oxford Classical Dictionary ("Names, Personal") and the article "What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700", in Journal of Roman Studies, vol. 84, pp. 124–145 (1994).
So interestingly, the Romans probably did refer to the name of a foreigner as nomen, but if he/she became a Roman citizen and his/her original name survived in the tria nomina (or maybe nomen+cognomen), it would be in the form of cognomen.