Contrary to what you suggest, Black Steel does not seem to be very special or specific in Westeros. It is fairly common and presumably numerous such weapons exist in the continent.
It has been used to make arms or pieces of armor. Following would be the examples:
AGOT, Tyrion I:
Clegane cast a long shadow across the hard-packed earth as his squire
lowered the black helm over his head1. "I could silence the
creature, if it please you," he said through his open visor.
AGOT, Eddard V:
Littlefinger led him into a tower, down a stair, across a small sunken
courtyard, and along a deserted corridor where empty suits of armor
stood sentinel along the walls. They were relics of the Targaryens,
black steel with dragon scales cresting their helms, now dusty and
forgotten. "This is not the way to my chambers," Ned said.
AGOT, Catelyn VII:
Bronn was so lightly armored he looked almost naked beside the knight.
He wore only a shirt of black oiled ringmail over boiled leather, a round steel halfhelm with a noseguard, and a mail coif.
AGOT, Eddard XIV:
But Littlefinger had kept his promise; all along the walls, in front
of Robert's tapestries with their scenes of hunt and battle, the
gold-cloaked ranks of the City Watch stood stiffly to attention,
each man's hand clasped around the haft of an eight-foot-long spear tipped in black iron. They outnumbered the Lannisters five to one.
AGOT, Tyrion VII:
Lord Lefford frowned. "I saw that great hairy one today, the one who
insisted that he must have two battle-axes, the heavy black steel
ones with twin crescent blades."
ACOK, Jon III:
"Give the wildling an axe, why not?" He pointed out Mormont's weapon,
a short-hafted battle-axe with gold scrollwork inlaid on the black
As evident, Black steel is shown to have been used in:
- Tips of spears
This appears to have been used from Targaryen Kings to a lowly sellsword like Bronn so it should be evident that Black steel is neither special nor very costly. However it must be noted that Bronn may have nicked his ringmail from some deadbody and sellswords usually carry most of their wealth in form of weapons, ornaments and steeds.
Not to mention, armors of Night's Watch are black, an order which is notoriously poor. So it shouldn't be very difficult or expensive to color your steel black.
As an afterthought, The song says that the singer died in a duel with the Dornishman so it is unlikely that the song is real at all. It is entirely possible however that this song maybe about exploits of someone else, written in that person's perspective.
As proven above, Black steel isn't something that special or unique that could be tied to some specific person. It can however be classified as some distinct sort of steel like Valyrian steel which tends to be heavier than usual steel.
I am not a metallurgy expert but I suspect that it is black because of higher proportion of crude iron (If the iron ore found in Westeros is Dark Grey in color. It must be since GRRM refers to Stannis as crude iron, black and brittle) and heavier because it is isn't beaten a lot into multiple layers, thus requiring more iron ore and resulting in a heavier product.
It is also possible that the steel is normal steel which is then enameled black or given black dye when forging into a weapon.
So to answer your question, It isn't possible to say that the Dornishman in question is some important or well-known figure. This might just be manifestation (Or for all we know, Origin) of the typical Westerosi stereotypes that Dornishmen are hot-headed fierce fighters while Dornishwomen are beautiful and wanton.
1. As one of our members objects that this quote only mentions a black helm which may have been made of candy instead of steel, I'd say that in the same chapter, it is specifically mentioned that it was the famous Hound-shaped helmet used by Sandor. I don't see him going to battle wearing a helm which is made of unicorn hair. I just thought it should be clear to everyone that it was steel. But anyhow, here goes from the same chapter:
It was fashioned in the likeness of a snarling black hound, fearsome
to behold, but Tyrion had always thought it a great improvement over
Clegane’s hideously burned face.
If that's still dubious for you, please refer to ACOK Tyrion XIII, where it is specifically referred to have been made of soot-colored steel.