Okay, an Elf has 4 names(Version 2):
1. Laura: Given to her by her mother.
2. Lisa: Given to her by the Elf that raised her.
3. Alexia: Given to her by a close friend (who was later discovered to be her father).
4. Svenethina: Given to her by one of the guards, whom she later marries.
 

Now, would Lisa still be her 2nd name, or would Alexia become her 2nd name?

Hmm, you have introduced a bit of a grey area here if you are wondering about 'true names' within the context of Middle-earth. In other words, at least in his essays on naming anyway, Tolkien does not account for a name given by an Elf who raised an Elf-child but who is not the father or mother of that child.

Basically what you are asking (it seems to me) is whether or not a Father-name -- at least in the sense of a name given by an Elf's actual father, and given to an Elf at some later point in time -- would necessarily supersede, in importance, a name already given by someone who is not actually the mother or father of the child, but raised the child.

Admittedly years later now! I can note that there are two main texts to consider when it comes to naming customs. Well, one would be too simple!

1) Morgoth's Ring: the account of naming among the Noldor

Here the 'true names' are the Father-name and the Chosen-name -- the Chosen-name concerns an Elf-child choosing his or her own name when deemed ready and capable of lamatyave, that is, of individual pleasure in the sounds and forms of words. The text suggests, at least, that this custom was specific to the Noldor, and not necessarily to the other Eldar.

The Father-name was public, the Chosen-name was private, but not secret. 

The Mother-name had authority, and was regarded as a true name when solemnly given, and was public if placed immediately after the Father-name. Mother-names of insight had a high position and in general use sometimes replaced both the Father-name and Chosen-name, though the Father-name and the Chosen-name remained ever the true or primary name, and a necessary part of any 'full title'.

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.

2) Note on Mother-names, The Shibboleth of Feanor

The second text is later than the one published in Morgoth's Ring. It's shorter, and concerns the Eldar in Valinor, not simply the Noldor. Here it is said that the Eldar had, as a rule, two names:

The Father-name, received at birth

The Mother-name, given later, often some years later, but sometimes soon after birth

Other kinds of names are noted, called the epesse of 'Aftername' and the Kilmessi 'Self-names', but it is said here that the 'true names' remained the Father and Mother names, although in later song and history any of the four might become the name generally used and recognized. The True names were not forgotten by the scribes and loremasters, or the poets.

Also the Noldor altered their Quenya names after returning to Middle-earth and adopting the Sindarin tongue.

It might also be noted that the Kilmessi or 'Self-names' in this text are not the same animal as the Chosen-names of the Noldor. The Kilmessi were connected with the idea that later some among the Exiles gave themselves names as disguises or in reference to their own deeds and personal history -- so these are different from the Elf-child's Chosen-name, and Christopher Tolkien even notes that the Chosen-name as described in the earlier work appears to have been abandoned here.

That may be so, but again in the earlier work the custom of the Chosen-name appears connected to the Noldor, and it may be that this briefer text is meant to concern the Eldar in general.

Again I'm not sure how these things might plug into the scenario above, but in any case we can see that according to both accounts, the Father-name appears to have a high status, so to speak.