I'm 100% sure that Lightbringer will not be any mere sword, but Dany's dragons. For a start, what makes more sense as a significant weapon against the Others - a fully-grown, trained dragon breathing rivers of fire, or one simple flaming sword? On that note...
"When your dragons were small they were a wonder. Grown, they are death and devastation, a flaming sword above the world." - Xaro Xhoan Daxos, A Dance With Dragons
There are multiple references in the text that equate the dragons with swords or weapons. And as we know from Maester Aemon, the prophecy text contains some translation errors (Prince/Princess) and a mistranslation of weapon to sword may simply be another.
But most importantly, the circumstances around the birthing of the dragons complies exactly with the prophecised forging of lightbringer.
"In ancient books of Asshai it is written that there will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world. In this dread hour a warrior shall draw from the fire a burning sword. And that sword shall be Lightbringer, the Red Sword of Heroes..." - Melisandre, A Clash of Kings (Davos I)
We have had this "day after a long summer" - autumn arrives at the end of AGOT/start of ACOK, at the same time as the red comet (or "Bleeding star" according to the Dothraki), on the day that Daenerys draws the dragons from the fire. It's also interesting that Melisandre seems to give the exact same set of circumstances for waking dragons from stone as she does for forging Lightbringer, yet never connects the two.
"When the red star bleeds and the darkness gathers, Azor Ahai shall be born again amidst smoke and salt to wake dragons out of stone." - Melisandre, A Storm of Swords (Davos III)
And then you have the parallels between Dany/Drogo and the story of Nissa Nissa. The original AA first failed to temper the sword, but succeeded when he plunged it into his wife's heart. This is probably not a literal story but an allegorical one - the main takeaway being that the forging of Lightbringer involved the sacrifice of a loved one, and obeys a rule we know well - only death can pay for life. Dany sacrifices her spouse in order to "forge" her dragons in the fire.
"It is said that her cry of anguish and ecstasy left a crack across the face of the moon, but her blood and her soul and her strength and her courage all went into the steel." - Salladhor Saan, A Clash of Kings (Davos I)
This is not directly related to the theory, but I found it interesting, if probably meaningless, that this passage uses the same "cracked moon" imagery that Doreah describes when talking about the original birth of dragons.
Related to this theory, though not part of it, is the idea that there is not one Azor Ahai but several - the three heads of the dragon, if you will.
"And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be Azor Ahai come again, and the darkness shall flee before him."
The prophecy is in fact telling us very clearly that they who hold Lightbringer, by simple virtue of holding it, are Azor Ahai. Anyone who tames and rides a dragon is therefore Azor Ahai. Three dragons, three Lightbringers, three dragonriders, three heads of the dragon. Three Azor Ahais.
tl;dr - Dany's dragons are Lightbringer, and if I'm wrong I'll swallow Dawn myself.