Word for word it's "Hello, what to-you is name". Literally meaning "what is the name (given) to you".
When saying "your/my name ..." it's common to use the dative form instead of the genitive. So instead of "meī/tuī nomen (my/your name)" it's "mihi/tibi nomen (name given to me/you)"
Nomen is the term used generically for name, which is why there is also the verb "nominare": to name or call by name. You are correct that a full Roman name has 3 (sometimes 4) parts, and nomen refers to the second part (praenomen the first, and cognomen the third), but in the context of asking for someone's name, generically, you would use "nomen".
Just sharing this found at http://www.cornelius93.com/CorneliaCoin-MainPage.html: in far ancient times Romans citizens possessed only one name like the most Indo-European people. It was during the Republican period, which began around 510 BC with the overthrow of the Etruscan monarchy and lasted roughly until around 44 BC when Julius Caesar was appointed as dictator, that a more stable naming system slowing emerged referred to as tria nomina. This is when most Roman Citizen took on three names. The first name was called the Praenomen and it designated the individual, similar to our first name. In ancient times the Romans had less than thirty first names and only about ten of them were common. To make the options even less, most clans favored certain Praenomina. The second, or middle, name was known as the Nomina and it referred to the nomen, or name, of the tribe. This was an inherited name shared by all the members of the family and even special slaves. Originally there were only three patrician tribes in ancient Rome; the Ramnes, Tities and Luceres. Within each of these tribes there were smaller clans. Later, about thirty or more 'plebeian tribes' were added. Over the years, due to war and other reasons, many clans disappeared. By the Middle Republican period a three-letter abbreviation for the tribe in which the man was enrolled was often used. The third name was called the Cognomen, or family name, similar to our last name. It was initially formed as stirps, or stems (lineages) off a given family to designate a line of descent of common ancestry. Although this naming behavior is rooted much earlier in Roman history, it would not begin to appear in public documents as commonplace until the time of Lucius Cornelius Sulla. As an example of what has just been said; Lucius Cornelius Sulla implies that Lucius, born within the gens of the Cornelius family tree, is from the stirps or lesser branch of the family known as Sulla. Another example is Publius Cornelius Scipio. His name means that Publius is of the gens Cornelius and the stirps Scipio. In both cases, being men, they assume the title Cornelius and not Cornelia. Sometimes a fourth, even fifth, name called Agnomen was added. In some cases, this name was a mark of great honor or a distinction which was carried after an outstanding exploit, such as a particularly successful military campaign. For example; Scipio received the honorary Agnomen 'Africanus' because of his military victories over Hannibal during the Second Punic War. Hence his full name became Publius Cornelius Scipio Africanus. In the case where a man was adopted into another family, which apparently happened with some frequency in ancient Rome, the individual would assume the Tria Nomina of his adopted family while adding the nomen of his birth gens at the end. An example of this behavior can be seen with Lucius Æmilius Paulus. He was adopted by Publius Cornelius Scipio and took the name Publius Cornelius Scipio Æmilianus. In a society as rigidly patriarchal as the Romans, women generally had no personal names and were known only by the feminine form of her tribe as reflected by the Nomina not the Cognomen. Thus, the daughter of Publius Cornelius Scipio was simply called Cornelia. In cases where more than one daughter existed in the same immediate family they would all have the same name but became designated by the order of birth. These names were kept even if they were married into a different clan. It would not be until the Imperial period of Rome when women were seen to commonly follow the Tria Nomina.
Quid tibi est nomen - I wonder if anyone can help me with:
Is nomen the subject of this sentence? As nomen is a neuter noun - is that why quid the neuter form of who is used? Why is the dative form of tu used - ie tibi. It doesn't appear to be working as a dative? Can anyone enlighten me? thank you
Yes, nomen is the subject. The interrogative pronoun quid is the complement so agrees with nomen in number, case and gender. Tibi is dative of possession, literally What to you is the name?
The dative of possession used with the verb to be emphasises the fact of possession, whereas the possessive adjective, meum nomen empasises the possessor. For example, if someone came into the room looking for David I might identify myself with Davidus nomen meum est. If, however, we're already talking and I'm giving my name I would say mihi nomen Davidus est.