Morgoth's Ring, 2nd paragraph, page 214. "Of Naming"
This is the manner in which the naming of children was achieved among the Noldor. Soon after birth the child was named. It was the right of the father to devise the first name, and he it was that announced it to the child's kindred upon either side. It was called, therefore, the father-name, and it stood first, if other names were afterwards added. It remained unaltered, for it lay not in the choice of the child.

But every child among the Noldor (in which point, maybe, they differed from the other Eldar) had also the right to name himself or herself. Now the first ceremony, the announcement of the father-name, was called the Essecarme or 'Name-making'. Later there was another ceremony called the Essecilme, or 'Name-choosing'. This took place at no fixed date after the Essecarme, but could not take place before the child was deemed ready and capable of lamatyave, as the Noldor called it: that is, of individual pleasure in the sounds and forms of words. The Noldor were of all the Eldar the swiftest in acquiring word-mastery; but even among them few before at least the seventh year had become fully aware of their own individual lalatyave, or had gained a complete mastery of the inherited language and its structure, so as to express this tyave skilfully within its limits. The Essecilme, therefore, the object of which was the expression of this personal characteristic, usually took place at or about the end of the tenth year.

Morgoth's Ring, 1st paragraph, p 215.
In elder times the 'Chosen Name', or second name, was usually freshly devised, and thought framed according to the structure of the language of the day, it often had no previous significance. In later ages, when there was a great abundance of names already in existence, it was more often selected from names that were known. But even so some modification of the old name might be made.

Now both these names, the father-name and the chosen-name, were 'true names', not nicknames; but the father-name was public, and the chosen name was private, especially when used alone. Private, not secret. The chosen names were regarded by the Noldor as part of their personal property, like (say) their rings, cups, or knives, or other posesessions which they could lend, or share with kindred and friends, but which could not be taken without leave. The use of the chosen name, except by members of the same house (parents, sisters, and brothers), was a token of closest intimacy and love, when permitted. It was, therefore, presumptuous or insulting to use it without permission. (This sentiment had thus nothing to do with 'magic' or with taboos, such as are found among Men.)

Morgoth's Ring, 3rd paragraph, p 215.
Since, however, the Eldar were by nature immortal within Arda, but were by no means changeless, after a time one might wish for a new name. (The Eldar hold that, apart from ill chances and the destruction of their bodies, they may in the course of their years each exercise and enjoy all the varied talents of their kind, whether of skill or of lore, though in different order and in different degrees. With such changes of 'mind-mood' or inwisti their lamatyaver might also change. But such changes or progressions were in fact seen most among the neri, for the nissi, even as they came sooner to maturity, remained then more steadfast and were less desirous of change. [According to the Eldar, the only 'character' of a person that was not subject to change was the difference of sex. For this they held to belong not only to the body (Hroa) but also to the mind (Indo) equally: that is, to the person as a whole. This person or individual they often called esse (that is 'name'), but it was also called erde, or 'singularity'. Those who returned from Mandos, therefore, after the death of their first body, returned always to the same name and to the same sex as formerly.]) He might then devise for himself a new chosen name. But this did not abrogate the former name, which remained part of the 'full title' of any Noldo: that is the sequence of all the names that had been acquired in the course of life.

Morgoth's Ring, 1st paragraph, p 216.
These deliberate changes of chosen name were not frequent. There was another source of the variety of names borne by any one of the eldar, which in the reading of their histories may to us seem bewildering. This was found in the Anessi: the given (or added) names. Of these the most important were the so-called 'mother-names'. Mothers often gave to their children special names of their own choosing. The most notable of these were the 'names of insight', essi tercenye, or of 'foresight', apacenye. In the hour of birth, or on some other occasion of moment, the mother might give a name to her child, indicating some dominant feature of its nature as perceived by her, or some foresight of its special fate. These names had authority, and were regarded as true names when solemnly given, and were public not private if placed (as was sometimes done) immediately after the father-name.

All other 'given names' were not true names, and indeed might not be recognized by the person to whom they were applied, unless they were actually adopted or self-given. Names, or nicknames, of this kind might be given by anyone, not necessarily by members of the same house or kin, in memory of some deed, or event, or in token of some marked feature of body or mind. They were seldom included in the 'full title', but when they were, because of their wide use and fame, they were set at the end in some form such as this: 'by some called Telcontar' (that is Strider); or 'sometimes known as Mormacil' (that is Blacksword).