The most powerful wand was called the 'Elder Wand'. But did the original wand Harry possess, made of holly and a phoenix feather core, have a name?
No, Harry and those around him never call his wand by a name.
Harry is the perspective character in a story told from a third-person limited perspective, meaning we know almost everything that goes on in Harry’s life and thoughts. Harry has never referred to his wand as anything other than his wand. Nor has anyone else named Harry’s wand for him.
In fact, the Elder Wand is the only wand with a name we know of.
As far as we know, none of the wizards or witches we see throughout has named their wands. Many of them are quite fond of their wands, but at least to our knowledge, none of them call their wands by any special name (like Doombringer, Shirley, or Wandy McWandface). They either refer to their wands as simply their wands, or by a descriptor like “my black walnut wand” especially if they’ve had more than one wand.
The Elder Wand was named because it’s a wand of ancient legend, and supposedly extraordinary power, so it’s a special case.
However, it’s entirely possible that some wizards do name their wands and we just never hear of it.
The elder wand is known by a few names
I sought a third wand, Severus. The Elder Wand, the Wand of Destiny, the Deathstick. Lord Voldemort
It seems wands are more generally described by their woods as is a more unique identifier than core or other qualities (38 woods, 21 lengths, 14 flexibility, 3 cores)
The rarest wand wood of all, and reputed to be deeply unlucky, the elder wand is trickier to master than any other. It contains powerful magic, but scorns to remain with any owner who is not the superior of his or her company; it takes a remarkable wizard to keep the elder wand for any length of time. The old superstition, ‘wand of elder, never prosper,’ has its basis in this fear of the wand, but in fact, the superstition is baseless, and those foolish wandmakers who refuse to work with elder do so more because they doubt they will be able to sell their products than from fear of working with this wood. The truth is that only a highly unusual person will find their perfect match in elder, and on the rare occasion when such a pairing occurs, I take it as certain that the witch or wizard in question is marked out for a special destiny. An additional fact that I have unearthed during my long years of study is that the owners of elder wands almost always feel a powerful affinity with those chosen by rowan.
While this does not imply the wand of oldest Peverell brother was the only wand made of elder (but it was supposedly the only wand made with Thestral hair), I would hazard a guess the legend behind it is what gave it the name.
No. But there are descriptions.
Harry's first wand was 11 inches long. This was described by Garrick Ollivander to be an unusual combination of wand core and wood. The feather was donated by Fawkes, Albus Dumbledore's phoenix. It was revealed by Garrick Ollivander that Tom Riddle's wand core also came from Fawkes, making the two wands "brothers". Harry's wand was described as being "nice and supple". http://harrypotter.wikia.com/wiki/Harry_Potter%27s_wand
Throughout the books, there aren't any mentioned information about what is the name of Harry's wand is. But some people name their wands and it's up to them. But it's not necessary to do so for the main reason that it's just a wand.
It's better for the Tale of the Three Brothers to have a name for the Elder wand.
The eldest brother, a combative man, asked for a wand more powerful than any in existence. Death granted his wish by fashioning the Elder Wand from a branch of a nearby elder tree standing on the banks of the river. TALE OF THE THREE BROTHERS **
The elder wand could have been made to emphasize the power of the most powerful wand. And it suits the wand better to have a name especially that the story is like a legend.
Elder brother's elder wand.
CORRECTION: as what @Skooba said, it's made from elder wood.