The following is how it happened in the books, with the links to the people/things mentioned there:
“...found one bastard,” one said. “The rest will come soon. A
day, two days, a fortnight...”
“And when he learns the truth, what will he do?” a second voice asked
in the liquid accents of the Free Cities.
“The gods alone know,” the first voice said. Arya could see a wisp of
grey smoke drifting up off the torch, writhing like a snake as it
rose. “The fools tried to kill his son, and what’s worse,
they made a mummer’s farce of it. He’s not a man to put that
aside. I warn you, the wolf and lion will soon be at each
other’s throats, whether we will it or no.”
“Too soon, too soon,” the voice with the accent complained. “What good
is war now? We are not ready. Delay.”
“As well bid me stop time. Do you take me for a wizard?”
The other chuckled. “No less.” Flames licked at the cold air. The tall
shadows were almost on top of her. An instant later the man holding
the torch climbed into her sight, his companion beside him. Arya crept
back away from the well, dropped to her stomach, and flattened herself
against the wall. She held her breath as the men reached the top of
“What would you have me do?” asked the torchbearer, a stout man in a
leather half cape. Even in heavy boots, his feet seemed to glide
soundlessly over the ground. A round scarred face and a stubble of
dark beard showed under his steel cap, and he wore mail over boiled
leather, and a dirk and shortsword at his belt. It seemed to Arya
there was something oddly familiar about him.
“If one Hand can die, why not a second?” replied the man
with the accent and the forked yellow beard. “You have danced the
dance before, my friend.” He was no one Arya had ever seen before, she
was certain of it. Grossly fat, yet he seemed to walk lightly,
carrying his weight on the balls of his feet as a water dancer might.
His rings glimmered in the torchlight, red-gold and pale silver,
crusted with rubies, sapphires, slitted yellow tiger eyes. Every
finger wore a ring; some had two.
“Before is not now, and this Hand is not the other,” the scarred man
said as they stepped out into the hall. Still as stone, Arya told
herself, quiet as a shadow. Blinded by the blaze of their own torch,
they did not see her pressed flat against the stone, only a few feet
“Perhaps so,” the forked beard replied, pausing to catch his breath
after the long climb. “Nonetheless, we must have time. The
princess is with child. The khal will not bestir
himself until his son is born. You know how they are, these savages.”
The man with the torch pushed at something. Arya heard a deep
rumbling. A huge slab of rock, red in the torchlight, slid down out of
the ceiling with a resounding crash that almost made her cry out.
Where the entry to the well had been was nothing but stone, solid and
“If he does not bestir himself soon, it may be too late,” the stout
man in the steel cap said. “This is no longer a game for two players,
if ever it was. Stannis Baratheon and Lysa Arryn have fled beyond my
reach, and the whispers say they are gathering swords around them. The
Knight of Flowers writes Highgarden, urging his lord father to send
his sister to court. The girl is a maid of fourteen, sweet and
beautiful and tractable, and Lord Renly and Ser Loras intend that
Robert should bed her, wed her, and make a new queen. Littlefinger...
the gods only know what game Littlefinger is playing. Yet Lord
Stark’s the one who troubles my sleep. He has the bastard, he has
the book, and soon enough he’ll have the truth. And now his wife
has abducted Tyrion Lannister, thanks to Littlefinger’s meddling. Lord
Tywin will take that for an outrage, and Jaime has a queer affection
for the Imp. If the Lannisters move north, that will bring the Tullys
in as well. Delay, you say. Make haste, I reply. Even the finest of
jugglers cannot keep a hundred balls in the air forever.”
“You are more than a juggler, old friend. You are a true sorcerer. All
I ask is that you work your magic awhile longer.” They started down
the hall in the direction Arya had come, past the room with the
“What I can do, I will,” the one with the torch said softly. “I must
have gold, and another fifty birds.”
She let them get a long way ahead, then went creeping after them.
Quiet as a shadow.
“So many?” The voices were fainter as the light dwindled ahead of her.
“The ones you need are hard to find... so young, to know their
letters... perhaps older... not die so easy...”
“No. The younger are safer... treat them gently if they kept their
tongues the risk...”
AGOT - Arya III
So the essence is, Illyrio and Varys were arguing about their plans to have the Targaryens strike west to push their claim with the help of the Dothraki.
Varys was concerned that Lord Stark had found one of King Robert's bastards, Gendry so no doubt he was headed on the same road as Lord Stannis and Lord Arryn had braved before. Sooner or later, he would find the other bastards as well and would see the same truth about parentage of King Robert's children with Cersei Lannister.
Illyrio was curious about Stark's possible reaction to the fact and Varys was convinced that due to the attempt on Bran's life, Starks and Lannisters would be at war soon. Illyrio then wanted him to delay the war until Targaryens were ready to invade, to which Varys replied in negative, saying that things were getting out of his hands.
Illyrio then suggested that if Jon Arryn could be murdered, why can't Lord Stark? That would delay the exposure of Robert's heirs' true parentage and delay the war between Great Houses of Westeros. Varys however did not believe that Eddard Stark could be murdered the same way.
Illyrio explained that because Daenerys was pregnant, Drogo would not invade until his child was born. That's why they needed to delay the civil war. Varys, impatient and frustrated, explains to him the recent developments. Stannis and Lysa Arryn were mustering soldiers, Loras Tyrell wanted Lord Mace Tyrell to send Margaery to the court, in hopes of getting Robert to set Cersei aside and wed her instead with aid of Lord Renly. Littlefinger was spinning his own web. And Lady Stark had kidnapped Tyrion Lannister, which was sure to incite a harsh and decisive reaction from the Rock.
So the murder plot in question, is a mere suggestion by Illyrio that they should remove Lord Stark from their path. To learn about the real deal with Varys and Illyrio, see my previous answer here (Spoilers alert).