While the Julii did not spawn quite as many magistrates during the era of Rome's Republic as did at least a dozen other notable gentes, the Julii nevertheless were one of the earliest and most distinguished families of ancient Rome. Livy (1.3) tells us how the Julii descended from and took their nomen gentilicium from Iulus, an alternate name of Ascanius, who was the son of the Trojan chief Aeneas.
As legend told that Aeneas was the son of Anchises and the goddess Venus, the Julii thereafter claimed divine descent from Venus -- an extraordinary claim which C. Julius Caesar (five times consul, 59, 48, 46, 45, and 44 BCE) was by no means reticent to use often and to his advantage for impressing both civilians and troops.
We know that the Julii were of Alban origin -- the town of Alba Longa, at the foot of Mount Alba, had not only been founded by Ascanius/Iulus, but Livy also tells us (1.30) that the Julii (along with the Servilii, Quinctii, Geganii, Curiatii, and Cloelii) were among the leading Alban families enrolled among the patricians by King Tullus Hostilius, following Rome's demolition of Alba.
I have attempted here to list and define the various surnames used by the Julii of the Republic, particularly those who served in magisterial positions during the time of the Republic as noted in Broughton's Magistrates of the Roman Republic. For the purpose of this list, I have included cognomina, adoptive cognomina, and agnomina under the collective term of "surnames."
Surnames of the Julii
Augustus - Meaning "majestic, venerable, worthy of honor," and originally a religious term. This titular surname was conferred upon Gaius Julius Caesar Octavianus by the Senate in 27 BCE, marking the close of Rome's period of the Republic and Rome's entry into its period of the Principate.
Bursio - Derived from the Greek word bursa, meaning "a hide." This cognomen of the Julii is known only from coins, and the individual Lucius Julius Bursio who bore this name was either a Questor or a Monetal, circa 84-83 BCE.
Caesar - A hereditary surname of the patrician Julii with several different, presumed meanings. The most likely is derived from the Latin word caesaries, meaning "hairy; possessing a head of hair" and possibly referring to the original Caesar having been born with a full head of hair. Pliny the Elder, in his Natural History (Book 7), is responsible for the enduring belief that the family derived their name from another Latin word implying that the first of the Caesones had been cut from his mother's womb (a story which is the source of the medical term "Caesarian section").
The earliest Caesar who served in a magisterial position (and who was also the first Caesar noted in history) was Sextus Julius Caesar, praetor in Sicily in 208 BCE.
Demetrias - Derived from the name of the Greek goddess of agriculture, Demeter (meaning "earth mother"), this foreign cognomen entered the Julian gens as the name of a freedman of Caesar (Gaius Julius Demetrias) who rose to a magisterial position (possibly that of prefect) in Cyprus, 39 BCE.
Helenus - Derived from the Greek helene and meaning "a shining torch, or corposant." This foreign cognomen entered the Julian gens as the name of a freedman of Octavianus (Gaius Julius Helenus) who rose to a magisteral position (possibly that of prefect or legate) in the Roman fleet and Sardinia in 40 BCE when he recovered Sardinia from the followers of Pompey.
Iullus, Iulus - See Jullus, Julus.
Jullus, Julus - An alternate name for Ascanius, the son of Aeneas and legendary founder of the Julia gens. Iulus/Julus is possibly a diminutive derivation of Iupiter/Jupiter, the Roman god with whom Ascanius was associated. This cognomen identified one of the oldest branches of the patrician Julia gens, as evidenced by the first Julian to become consul, in 489 BCE, Gaius Julius Jullus (or, Gaius Iulius Iullus).
Libo - Meaning "sprinkler," a surname possibly derived from the office of libation-pourer at the sacrifice. This cognomen was also found in the gentes of the Marcii, Poetelii, Scribonii, and Statilii.
Mento - Derived from the Latin word mentum, meaning "chin," this surname referred to a person possessing a long, or prominent chin.
Mocilla - This otherwise unknown surname of the Julii may actually be an error for the surname "Ocella" (meaning "having small eyes") of the Livii. Lucius Julius Mocilla was recorded by the Roman biographer Cornelius Nepo (in Atticus) as having been a staunch republican and praetorian who fought in the army of Cassius and Brutus at Philippi, and who afterwards was protected along with his son by Titus Pomponius Atticus, who sent Mocilla supplies from Epirus to Samothrace. However, classicists such as Friedrich Münzer and Conrad Cichorius identify this Mocilla with Lucius Livius Ocella.
Octavianus - An adoptive surname which found its way into the Julia gens when the testamentary adoption of Gaius Octavius by Caesar was confirmed by a curiate law. The gens name of Octavius is derived from the Latin octavus, meaning "eighth," a name that would have been given to the eighth-born son in a Roman family.
Salinator - Meaning "salt-dealer," the only instance of this surname appearing in the Julia gens being found in the name of Lucius Julius Salinator, a legate who served under Sertorius in Spain in 81 BCE (as recorded by Plutarch in Sertorius). However, as with the surname of Mocilla (see above), classicists such as Conrad Cichorius are of the opinion that once again a member of the Julii has been confused with a member of the Livii in manuscript translations, and that this Salinator was actually Lucius Livius Salinator.
Strabo - Meaning "squinter; squint-eyed," and referring to a defect of the eyes and vision. This cognomen was also found in other gentes, such as the Fanii and the Pompeii, and the medical term "strabismus" today refers to a condition of misalignment of the eyes that prevents proper binocular vision.
Vopiscus - Originally a rare praenomen used in the Julia gens, this name was also used as an occasional additional cognomen by Gaius Julius Caesar Strabo Vopiscus (quaestor circa 96 BCE, and later pontifex). The name "Vopiscus" refers to one of two twins -- the one who survives in the womb to full term, when the other twin is miscarried.
References: Broughton, T. Robert S.The Magistrates of the Roman Republic. 2 vols.New York: The American Philological Association, 1952. Chase, George Davis. "The Origin of the Roman Praenomina." Harvard Studies in Classical Philology, Vol. 8. (1897), pp. 103-184. Cichorius, Conrad. Römische Studien: Historisches, Epigraphisches, Literaturegeschichtliches aus vier Jahrhunderten Roms. Leipzig-Berlin: B.G. Teubner, 1922. Justin, Cornelius Nepo and Eutropius. Trans. John Selby Watson. London: Henry G. Bohn, 1853. Lewis, Charlton T. and Charles Short. A Latin Dictionary: Founded on Andrews' Edition of Freund's Latin Dictionary. Oxford: Clarendon Press, 1879. Livius, Titus. The History of Rome: The First Eight Books. Trans. D. Spillan. London: Henry G. Bohn, 1853. Livius, Titus. The History of Rome, Books Twenty-seven to Thirty-six. Trans. Cyrus Edmonds. London: Henry G. Bohn, 1850. Pliny the Elder [Gaius Plinius Secundus]. Natural History. Trans. John Bostock and Henry Thomas Riley. London: Taylor and Francis, 1855. Plutarch. Sertorius. Trans. John Dryden. Provided by The Internet Classics Archive.http://classics.mit.edu/Plutarch/sertoriu.html Smith, William, ed. Dictionary of Greek and Roman Biography and Mythology. London: John Murray, 1890.
There was no direct Roman equivalent of "sir" or "madam".