Summarized from: Of the Laws and Customs Among the Eldar, Morgoths Ring pp 214-217 by Ithildin
This essay (penned by JRRT possibly in the late 1950s) pertains specifically to Noldorin customs, though the wording suggests that the customs may possibly extend to other Eldar with the exception of Essecilmė, the chosen name. I highly recommend the entire essay, as it deals with many other aspects of elvish belief and custom in addition to those I have attempted to summarize here.
This ceremony took place soon after the childs birth, and in it the father announced the name he had devised for the child to kin on both sides of the family. This name was considered the childs true name and remained unchanged throughout life (except as the language itself evolved and changed over the ages).
It seems that this name remained the persons official name even when other names replaced it in general use.
Essecilmė Name-Choosing (apparently primarily a Noldorin custom)
Noldor children had the option of choosing a name for themselves when they achieved mastery of the language. The ceremony could not take place before the child was deemed ready and capable of lįmatyįvė, as the Noldor called it: that is the individual pleasure in the sounds and forms of words.
This seldom occurred before the age of 7 and usually Essecilmė ceremonies occurred around the end of the 10th year.
This name expressed some personal characteristic of the person, something more individual than physical features. Though rarely done, an additional name could be chosen at a later date if desired. Each name became part of the Noldos full title, the sequence of all the names that had been acquired in the course of life.
The chosen name was considered a true name (as opposed to a nickname) but unlike the father-name, the chosen name would be considered private property and would be used only by immediate family and the closest of friends. It would be considered presumptuous or insulting for someone to use it without permission.
In the early ages, these names were usually freshly devised however as ages went on (and more names were available) they were more often chosen from existing names, though the name might be modified in some way. Essecilmė ceremonies were never frequent and became more rare as the ages wore on.
Anessi Given (added) Names
The most important of the anessi would be the mother-names: The amilessi tercenyė (mother-names of insight) and of foresight, amilessi apacenyė. These were bestowed upon the child when the mother perceived by insight some dominant feature of the childs nature; or if through foresight she perceived the child was destined for some special fate. The essi tercenyė (names of insight) were highly regarded, and sometimes replaced the father-name and chosen name in general use, though the father-name (and chosen name when applicable) were always considered the true/official name(s). These were public names if placed after the father-name in the full title. The amilessi tercenyė were more common (and more often came into public use) in the early years of the Eldar when father-names were usually a modification of the fathers name.
Other anessi could be given by anyone, and might not even be recognized by the person unless they were adopted or self-given. They pertained to some deed or event or some prominent feature of body or mind. These given names or nicknames were not considered true names, though some achieved wide general use. They were seldom included in the full title and if by wide general use they were, they came at the end.
Fėanor (son of Finwė):
Father-name: Finwion (later modified to: Curufinwė when his talent was revealed)*
amilessi tercenyė (mother name of insight) Feanįro Spirit of Fire
Chosen name: Feanįro (in honor of his mother whom he never saw) **
Fėanor = blend of Quenya Feanįro and Sindarin Faenor
Elwė lord of the Teleri
Anessė (given name): Sindicollo Grey-cloak
Sindarin form: Elu Thingol
Commonly called Thingol by others,
However, Elu or Elu-thingol would have been the proper title in his own realm.
* This modification seems to conflict with the earlier statement that the father-name remained unchanged throughout life.
** According to this essay, apparently it was reported but not known for certain if Fėanor in fact took his amilessė tercenyė as his chosen name.