In the episode Garden of Bones (season 2, episode 4), Catelyn sides with Renly instead of Stannis even though Ned supported Stannis.
Extract video here.
Stannis: "It is strange to find you beside my brother, Lady Stark. Your husband was a supporter of my claim. Lord Eddard’s integrity cost him his head. And you sit beside this pretender and chastise me."
And then Lady Stark doesn't really answer and changes the subject.
They knew Renly's host would a be threat to Lannister power in the Riverlands and possibly Casterly Rock.
In A Clash of Kings while Robb is in Riverrun they learn that Ser Stafford Lannister is gather another host at Casterly Rock. Ser Brynden Tully informs Catelyn of this prior to informing Robb
... "Lord Tywin is not the Kingslayer. He will not rush in heedless. He will wait patiently for Ser Stafford to march before he stirs from behind the walls of Harrenhal."
"Unless..." sais Catelyn.
"Yes?" Ser Brynden prompted.
"Unless he must leave Harrenhal," she said, "to face some other threat."
Her Uncle looked at her thoughtfully. "Lord Renly."
"King Renly." If she would ask help from the man, she would need to grant him the style he had claimed for himself.
"Perhaps." The Blackfish smiled a dangerous smile. "He'll want something, though."
"He'll want what kings always want," she said. "Homage."
A Clash of Kings, Chapter 7
It is my thought that they chose Renly to go to first for two reasons:
However, none of that really mattered when Stannis decided to lay siege to Renly's host in Chapter 22.
Catelyn admits to Stannis that she is not there to take sides between Renly and Stannis but needs the overall help of the Baratheons if Robb is to win his war.
... Stannis studied her face. "And what cause brings you to the field, my lady? Has House Stark cast its lot with my brother, is that the way of it?"
This one will never bend, she thought, yet she must try nonetheless. Too much was at stake. "My son reigns as King in the North, by the will of our lords and people. He bends the knee to no man, but hold out the hand of friendship to all."
A Clash of Kings, Chapter 31
Skooba has added a good answer but there is one thing that he has not mentioned from the book's perspective. Since you have added the books tag, I suppose I should mention it.
In Books, Northerners didn't know (At that time) the whole story about Joffrey being a bastard. For them, both Stannis and Renly were pretenders, out there to usurp their nephew's throne.
For them the choices were:
They were considering their options in AGOT Catelyn IX:
Lord Jonos Bracken rose to insist they ought pledge their fealty to King Renly, and move south to join their might to his.
“Renly is not the king,” Robb said. It was the first time her son had spoken. Like his father, he knew how to listen.
“You cannot mean to hold to Joffrey, my lord,” Galbart Glover said. “He put your father to death.”
“That makes him evil,” Robb replied. “I do not know that it makes Renly king. Joffrey is still Robert’s eldest trueborn son, so the throne is rightfully his by all the laws of the realm. Were he to die, and I mean to see that he does, he has a younger brother. Tommen is next in line after Joffrey.”
"Tommen is no less a Lannister," Ser Marq Piper snapped.
"As you say," said Robb, troubled. "Yet if neither one is king, still, how could it be Lord Renly? He's Robert's younger brother. Bran can't be Lord of Winterfell before me, and Renly can't be king before Lord Stannis."
"Renly is crowned," said Marq Piper. "Highgarden and Storm's End support his claim, and the Dornishmen will not be laggardly. If Winterfell and Riverrun add their strength to his, he will have five of the seven great houses behind him. Six, if the Arryns bestir themselves! Six against the Rock! My lords, within the year, we will have all their heads on pikes, the queen and the boy king, Lord Tywin, the Imp, the Kingslayer, Ser Kevan, all of them! That is what we shall win if we join with King Renly. What does Lord Stannis have against that, that we should cast it all aside?"
“The right,” said Robb stubbornly. Catelyn thought he sounded eerily like his father as he said it.
So until that point, Robb was predisposed towards Lord Stannis instead of Renly. It was Great Jon Umber's Proclamation which changed everything. Quoting from the same chapter:
“MY LORDS!” he shouted, his voice booming off the rafters. “Here is what I say to these two kings!” He spat. “Renly Baratheon is nothing to me, nor Stannis neither. Why should they rule over me and mine, from some flowery seat in Highgarden or Dorne? What do they know of the Wall or the wolfswood or the barrows of the First Men? Even their gods are wrong. The Others take the Lannisters too, I’ve had a bellyful of them.” He reached back over his shoulder and drew his immense two-handed greatsword. “Why shouldn’t we rule ourselves again? It was the dragons we married, and the dragons are all dead!” He pointed at Robb with the blade. “There sits the only king I mean to bow my knee to, m’lords,” he thundered. “The King in the North!”
Now the Northmen were not looking for an Overlord, who had to have a good claim to the throne. Now they were independent, negotiating as equals and had to find an ally who could help them. The logical choice was King Renly who had Stormlands and the Reach behind him.
Skooba has already correctly pointed out that It was Catelyn who knew what to do when news of another Lannister Army reached them.
Not to mention, The quote you mentioned is a show only thing. In Books, Stannis never knew that Lord Eddard was in favor of his claim.
As already mentioned by Skooba, this is how it went:
Stannis studied her face. “And what cause brings you to this field, my lady? Has House Stark cast its lot with my brother, is that the way of it?”
This one will never bend, she thought, yet she must try nonetheless. Too much was at stake. “My son reigns as King in the North, by the will of our lords and people. He bends the knee to no man, but holds out the hand of friendship to all.”
Catelyn didn't stay quiet. She answered Stannis frankly and showed that she and her lot did not know about the dark truth behind Joffrey's parentage:
Stannis frowned at her. “You presume too much, Lady Stark. I am the rightful king, and your son no less a traitor than my brother here. His day will come as well.”
The naked threat fanned her fury. “You are very free to name others traitor and usurper, my lord, yet how are you any different? You say you alone are the rightful king, yet it seems to me that Robert had two sons. By all the laws of the Seven Kingdoms, Prince Joffrey is his rightful heir, and Tommen after him... and we are all traitors, however good our reasons.”
ACOK- Catelyn III
