No, there's no especial magical power behind the names of things in the Tolkien Legendarium. It's impossible to prove this negative, so you're just going to have to trust me on this.
There can be significance in invoking the names of certain individuals, or giving (and revoking) names to individuals based on circumstance:
The most prominent example is in the case of Elbereth (another name for the Vala Varda). Her name is occasionally used as a charm against evil powers; as Frodo does on Weathertop:
Two of the figures halted. The third was taller than the others: his hair was long and gleaming and on his helm was a crown. In one hand he held a long sword, and in the other a knife; both the knife and the hand that held it glowed with a pale light. He sprang forward and bore down on Frodo.
At that moment Frodo threw himself forward on the ground, and he heard himself crying aloud: O Elbereth! Gilthoniel! At the same time he struck at the feet of his enemy. A shrill cry rang out in the night; and he felt a pain like a dart of poisoned ice pierce his left shoulder.
Fellowship of the Ring Book I Chapter 11: "A Knife in the Dark"
As discussed in dlanod's answer here, though, that invocation doesn't seem to have any magical effect; against the ringwraiths it acts more as a "don't mess with me or my big brother Elf will beat you up" declaration
Melkor is given a new name after the theft of the Silmarils and slaying of Finwë, signifying his new position in Elvish cosmology:
Fëanor rose, and lifting up his hand before Manwë he cursed Melkor, naming him Morgoth, the Black Foe of the World; and by that name only was he known to the Eldar ever after.
The Silmarillion III Quenta Silmarillion Chapter 9: "Of the Flight of the Noldor"
Thorin was given an epithet, "Oakenshield," to commemorate a feat of courage in the Battle of Azanulbizar:
It is said that Thorin's shield was cloven and he cast it away and he hewed off with his axe a branch of an oak and held it in his left hand to ward off the strokes of his foes, or to wield as a club. In this way he got his name.
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
Sam's surname changed from Gamgee to Gardener in the years following the War of the Ring, signifying his elevation from servant to gentry
In this particular case, Aragorn is just listing his credentials, albeit taking his sweet time about it:
Elendil! is his battle-cry; consider in Moria, for instance:
With a bound the Balrog leaped full upon the bridge. Its whip whirled and hissed.
'He cannot stand alone!' cried Aragorn suddenly and ran back along the bridge. 'Elendil!' he shouted. 'I am with you, Gandalf!'
Fellowship of the Ring Book II Chapter 5: "The Bridge of Khazad-dûm"
In this context, it presumably means "Listen to me!"1
Aragorn son of Arathorn is, of course, his given name and lineage. Identifying yourself as "X, son/daughter of Y" is quite common in civilizations without surnames2
Elessar/Elfstone is his regnal name, and may have a certain mythological significance, as Galadriel says:
'This stone I gave to Celebrían my daughter, and she to hers; and now it comes to you as a token of hope. In this hour take the name that was foretold for you, Elessar, the Elfstone of the house of Elendil!'
Fellowship of the Ring Book II Chapter 8: "Farewell to Lórien"
"Elfstone" is just "Elessar" translated into Westron; presumably for the benefit of anyone present who doesn't speak Elvish. He presumably announces it here to indicate that "Aragorn, son of Arathorn" (who most people have probably not heard of) is a Very Important Individual, and friendly with the Elves
Dúnadan is the singular of Dúnadain, identifying him as one of that folk. He's presumably proud of his history, but also marks him as being a descendant of the North Kingdom
Heir of Isildur Elendil's son is self-explanatory, and perhaps a bit redundant ("Isildur? Which Isildur are you heir of? Ohhh, that Isildur!")
The Sword that was Broken and is forged again is his evidence of the previous
Although very drawn-out, it seems reminiscent of the tendency among royalty in our world to announce themselves by extensive titles and epithets. Consider Queen Elizabeth, whose full title is so convoluted, it has its own Wikipedia page.
This isn't entirely sprung from my own head; it's worth noting that the reactions of Legolas, Gimli, and Éomer are essentially "okay, this guy is very important; I should probably kneel before him now":
Gimli and Legolas looked at their companion in amazement, for they had not seen him in this mood before. He seemed to have grown in stature while Éomer had shrunk; and in his living face they caught a brief vision of the power and majesty of the kings of stone. For a moment it seemed to the eyes of Legolas that a white flame flickered on the brows of Aragorn like a shining crown.
Éomer stepped back and a look of awe was in his face. He cast down his proud eyes. 'These are indeed strange days,' he muttered. 'Dreams and legends spring to life out of the grass.
The Two Towers Book III Chapter 2: "The Riders of Rohan"
Phyneas (obliquely) reminds me in comments that there is one occasion where it's possible a name is used as a magical invocation; Sam invokes the name of Elbereth to get past the Watchers, to escape from the tower or Cirith Ungol:
It's the gate. There's some devilry there. But I got through, and I'm going to get out. It can't be more dangerous than before. Now for it!'
Sam drew out the elven-glass of Galadriel again. As if to do honour to his hardihood, and to grace with splendour his faithful brown hobbit-hand that had done such deeds, the phial blazed forth suddenly, so that all the shadowy court was lit with a dazzling radiance like lightning; but it remained steady and did not pass.
'Gilthoniel, A Elbereth!' Sam cried. For, why he did not know, his thought sprang back suddenly to the Elves in the Shire, and the song that drove away the Black Rider in the trees.
'Aiya elenion ancalima!' cried Frodo once again behind him.
The will of the Watchers was broken with a suddenness like the snapping of a cord, and Frodo and Sam stumbled forward.
Return of the King Book VI Chapter 1: "The Tower of Cirith Ungol"
However, it's not clear whether the invocation was related to the magical effect. On one hand, the sequence of events indicates that the will of the Watchers didn't break until after Sam and Frodo invoked Elbereth. On the other hand, Sam previously used the Phial to get past the Watchers (and past Shelob, earlier) without needing any invocation. So it's not clear.
1 I'm not apologizing for this.
2 Note that, because of the in-universe conceit of The Lord of the Rings being translated from Westron into English, it's entirely possible that Aragorn is actually saying something like "Aragorn Arathornson", or "Aragorn ibn Arathorn" and Tolkien is helpfully translating that into something more accessible. I don't think there's any evidence for this (Westron wasn't a very well-developed language, relatively speaking), but I quite like it as a headcanon