Wikipedia says his name translate to "Fasting Coyote", does this mean that the nahuas fasted when someone died?, was fasting an act of mourning?, if so, wouldn't it be more accurate to translate it as "Mourning Coyote"?
Good question. In my research I've only ever come across Nezahualcóyotl as his birth-name, which is usually translated as “Fasted Coyote”. So the short story is that according to the sources the Acolhua ruler was given the name at birth and didn't change it upon the assassination of his father Ixtlilxochitl. But let's see what the sources have to say about it.
According to the Spanish wiki which I'm guessing you meant, the ruler was given the name Acolmiztli at birth “but the sad circumstances surrounding his adolescence [meaning his father's death] led to his name being changed to Nezahualcóyotl meaning “coyote that fasts or hungry coyote”, with the fast understood as a form of sacrifice” [my transl.]. No citation is given, so not a good sign. I first want to look at an example from an early colonial codex and than at a passage from the colonial native author Fernando de Alva Ixtlilxochitl for this – again, both give the name at his birth.
A word on name glyphs: Since no pre-Hispanic codices from the Valley of Mexico have come down to us, quite some knowledge of how to interpret Nahua glyphs was lost. So that there is uncertainty surrounding the meaning of some glyphs, that usually represent the names of places or rulers. Luckily this isn't the case for Nezahualcóyotl. Jongsoo Lee has written a great book on the ruler and the myths surrounding him called “The Allure of Nezahualcoyotl”. In it he discusses one native historical annal, the Codex en Cruz from early colonial times. The Codex gives the future ruler's birth date at 1 Tochtli (1 Rabbit) or 1402 as the most significant event in Tezcoco, the Acolhua capital. So above Tezcoco is a cradle connected with the name glyph of the ruler: a coyote with a collar. Lee (p. 257) translates the glyph as
Nezahualcoyotl [nezahual(-li) (fasted) + coyotl (coyote)]. “Fasted Coyote”
I couldn't find the right Codex en Cruz image online, but these two give a good impression of the glyph in question: This redrawing from the Codex Xolotl is quite schematic, but shows well the coyote head and the collar above the ruler, who is seated on a mat to show his importance as a tlatoani (ruler). This one from the Codex Telleriano-Remensis shows the original in color, and identifies both parts (the second glyph there shows the ruler's son and next ruler Nezahualpilli. So here we can see the name given clearly already at birth in an early colonial codex, plus the most common translation. Let's see what an Acolhua chronicler writing ca. 100 hears after conquest had to say about it.
Fernando de Alva Ixtlilxochitl was a noble historian who descended directly from the Acolhua rulers of Tezcoco – he was a descendant of Nezahualcóyotl. His work is also our main source for the pre-Hispanic Acolhua. To make a longer story short, this means that Alva Ixtlilxochitl had a big interest in portraying the Acolhua in a positive light, because his own family's property rights depended on this descendence.
I'll have a short look at his best-known chronicle, the Historia de la nación chichimeca (ca. 1625), where Nezahualcóyotl plays the most important part. For Alva Ixtlilxochitl his ancestor was among other things a cultured poet, a lawgiver and, in a clear fabrication, a proto-Christian. Coming back to your question, according to his descendant Nezahualcóyotl's father huehue (the elder) Ixtlilxochitl had two children: and “the first was the prince Acolmitzli Nezahualcóyotzin” [my transl., below as well]. The -tzin part here is a common honorific in Nahuatl, here used for the tlatoque. The first name according to Lee translates as “Arm of a Lion”, presumably referring to strength. So here we have both names together at birth – in contrast to the wikipedia article mentioning only Acolmitzli as his birth name.
For the second part of your question we'll turn to Alva Ixtlilxochitl's description of huehue Ixtlilxochitl's death – according to wiki the source for the presumed name change. First some context: In the early 15th c. the Tepanec ruler Tezozómoc had asserted dominance over the Valley of Mexico, in part by having the Acolhua ruler Ixtlilxochitl killed. After a longer exile the Mexica (aka known as the Aztecs) would overthrow the Tepanec and form the Aztec Triple Alliance: together with the Acolhua under Ixtlilxochitl's son Nezahualcóyotl, and the Tepanec of Tlacopan. This alliance under the Mexica in turn was dominant until the Spaniards' arrival. So we have here one crucial moment in Acolhua history.
Alva Ixtlilxochitl tells us how in this event Ixtlilxochitl was strongly wounded, and sent his soldiers away before Tezozómoc's last assault on his life. Here he addresses his son for the last time:
My beloved son, arm of the lion, Nezahualcóyotl … I charge you with this, not to forsake your subjects and vasalls, and not to forget that you are chichimeca, to recover your empire, … and to avenge your father's death
This is highly charged scene, where the author makes connections between the earlier chichimeca and the Acolhua in order to make his ancestors appear more powerful. More importantly for us, I'd argue that this isn't only a scene of mourning between father and son, but the focus is more on revenge for the (according to the author) illegitimate takeover of the Acolhua realm. This vengeance also provides the narratives' centre, with the 15-year old Nezahualcóyotl going to exile for years and later reconquering his father's realm. The other point is that once more, the later ruler is addressed here as “arm of the lion, Nezahualcóyotl” and at a later point as “Acolmitzli Nezahualcóyotl” – we have a parallel with his birth names, and no mention of a new name taken up.
The last thing to address for your questions is the name's meaning. Regarding the “fasting” part, I'd say that the fast had a big religious importance for the Nahua. This meant especially priests fasting before major ceremonies like the New Fire ceremony (every 52 years), but also before festivals like for main deities like Tezcatlipoca or Huitzilopoctli has been noted. Apparently even rulers would fast in specific periods. Actually Alva Ixtlilxochitl has a passage where (iirc) Nezahualcóyotl fasts for a longer period after the tragic passing of one of his wives, and during this time encounters a deity. So we can note the religious/cosmological importance of fasting, and I haven't come across fasting as a form of mourning although I wouldn't exclude it. As to the name's coyote part, I haven't come across a clear explanation for it. I could imagine that it's tied to Nezahualcóyotl birth date, and/or as form of reverence to the deity Huehuecóyotl, but would have to look more into it.
I'll end on a side note that goes beyond the question: Lee's book that I mentioned is very interesting in that it actually questions the central position of Nezahualcóyotl in Mexican culture until today. In short, his argument goes that later depictions of the Nahua ruler mostly build on colonial writer's, esp. on Alva Ixtlilxochitl and on the Fransciscan Torquemada who worked together. This meant for Lee that until today scholars and writer's haven taken up an Acolhua bias that construes Nezahualcóyotl as an exceptional poet and proto-Christian, and Tezcoco as a non-violent center of civilisation – wheareas in reality the Acolhua were not all that different from the other Nahua groups of the time. So yeah, a long way of saying I should probably edit the wiki article when I have the time.