I am not actually asking about who gave Daenerys the dragon eggs she receives in the 1st season of Game of Thrones. My question is, what is the actual history of Daenerys' dragon eggs?
At this point in time we know that Magister Illyrio Mopatis obtained them in his trading in the Shadow Lands beyond Asshai.
"What are they?" she asked, her voice hushed and full of wonder.
"Dragon's eggs, from the Shadow Lands beyond Asshai," said Magister Illyrio. "The eons have turned them to stone, yet still they burn bright with beauty."
A Game of Thrones, Chapter 11
The show repeats the same line (at about 3:25)
To move beyond the show and even more into the ASoIaF series and the accompanying materials, we have two popular theories to consider.
That Varys and/or Illyrio are secretly descended from the Blackfyre house.
The eggs given to Daenerys were somehow recovered from the Tragedy at Summerhall.
These are only theories (and TBH, I do not even fully believe the second theory), and the show doesn't seem to be headed in that direction as Daenerys is already on her way across the Narrow Sea and
the character of Aegon Targaryen (son of Elia and Rhaegar) was not introduced.
So it seems that the novels and show will differ on this topic, and so as stated above, for the purpose of the show, the eggs simply came from the Shadowlands.
In the books, there is a point of history around the time Rhaegar Targaryen was born. It had been a few years since the last dragon had passed away and their sizes were comparable to those of cats.
Under pressure to hold the power, the Targaryens tried to hatch the last three remaining dragons at a castle called Summerhall. They tried hatching them using wildfire and the flame that erupted was out of control resulting in the destruction of the castle and death of Aegon V, Prince Duncan and Ser Duncan the Tall. The eggs were believed to be lost in the process.
Since the show does not cover the past in such detail, these facts are skipped. The three dragon eggs Dany is offered are believed to be the ones obtained from Summerhall by unknown people and later sold them off to the Merchant.
PS: The proximity of the event in time to Rhaegar's birth was a reason they believed he was the Prince That Was Promised as he is said to be born under a bleeding star.
Find more here.
There is some good information on dragons in GOT in this source. It explains:
Five thousand years ago, men of the Valyrian Freehold learned how to master and ride dragons as beasts of war, and used them to forge an empire that stretched across most of the continent of Essos, dominating almost half of the Known World. Four hundred years before the War of the Five Kings, the entire Valyrian empire and almost all of its dragons were destroyed in a single day, during a cataclysmic volcanic eruption known as the Doom of Valyria. One Valyrian noble family, the Targaryens, survived the Doom on the distant island outpost of Dragonstone in the Narrow Sea - along with the last surviving Valyrian dragons.
In the series the dragon species is established as having been extinct for many years. The only thing left of the race are petrified dragon eggs, which are used as decoration, and the bones which are used for weapon crafting. According to one legend, dragons originated from a second moon that hatched when it drifted too close to the sun. This is mostly dismissed as a simple myth.