I think Eddard Stark with knowledge of twincest would be far more dangerous, compared to possible reaction of North for a dead Eddard Stark. Especially when his heir was a 'kid' (Robb). Sending him to the Wall would not solve the problem because, as we see with Benjen Stark; he could still visit Winterfell and tell everyone what he knows.
It's very important that Cersei didn't only send Eddard to the Wall; she required him to publicly confess to treason. On the steps of the Great Sept, Eddard told the crowd that he had tried to seize the throne for himself, and Joffrey was the legitimate son of Cersei and King Robert.
If Eddard had gone to the Wall as planned, but later spoken out against Cersei, he would have two major problems:
- Brothers of the Night's Watch are not permitted to take part in the politics of the Seven Kingdoms. If Eddard opposed Cersei, he would be breaking his vows and endangering the NW as a whole (because it depends for its long-term survival on neutrality in any conflicts which take place in the south). If necessary, the other NW brothers might have taken steps to stop him from interfering in politics.
- Eddard would effectively be saying that his solemn confession of treason was a lie told to save his own life. Cersei could then say the confession was true, and Eddard was telling lies because he still wanted the Iron Throne for himself. Either way, Eddard was lying. So his reputation as an extremely honourable man is ruined, which would greatly reduce his authority.
The other reason Cersei wanted Eddard to live, is that the Lannisters were facing a civil war against Renly and Stannis. If Eddard was allowed to join the Night's Watch, Robb Stark might be persuaded to remain neutral in the war, allowing the Lannisters to deal with their other enemies. If instead Eddard was executed, Robb and the other northern lords would almost certainly go to war.
As it turns out,
Eddard is killed and this is exactly what happens. Eddard was a very popular leader in the North, so the northern lords were almost as angry as Robb, and more than willing to support a war against the Lannnisters.
So allowing Eddard to live was a calculated risk, but one Cersei was willing to take.
Royal Canadian Bandit's answer is pretty much it. I will however expand on that.
Why Cersei needed Eddard Stark's cooperation?
Varys explained why Cersei needed Eddard Stark in AGOT:
“Cersei is frightened of you, my lord... but she has other enemies she fears even more. Her beloved Jaime is fighting the river lords even now. Lysa Arryn sits in the Eyrie, ringed in stone and steel, and there is no love lost between her and the queen. In Dorne, the Martells still brood on the murder of Princess Elia and her babes. And now your son marches down the Neck with a northern host at his back.”
So Cersei needed Eddard's cooperation in appeasing the North because:
- Tullys had tied up Lannister armies in Riverlands.
- Lady Arryn had immense strength and was sitting quietly in Vale without showing any signs of which side she might take. She was Lord Tully's daughter and Lord Eddard's sister in law so naturally Cersei feared she might send her armies to Riverlands against the Lannisters. Both women did not like each other.
- Prince Doran Martell of Dorne had a blood feud with Lannisters due to murder of Elia Martell and her Children.
- To make things worse, Robb Stark was marching with his bannermen to take the fight to Lannisters in order to win release of his father.
Rationale for Cersei's offer
Cersei was smart enough to realize she had no need to fight the Northmen if a compromise could be reached with Eddard Stark. Her biggest concern was claims of Robert's brothers. She knew if she could make her peace with Eddard Stark, she would be able to accomplish the following:
- War in Riverlands would cease and Lannister Armies would be free to mobilize against Baratheons. Lannisters attacked Riverlands only because of Tyrion's abduction and their marital alliance with North.
- Lady Arryn would have no reason to rise up against Cersei as Riverlands would be at peace then and North would also be at peace with Lannisters.
- Robb Stark could be persuaded to go back to Winterfell as his objective was to save his father from execution.
Cersei wanted to conserve power of house Lannister to fight against real threats of Renly and Stannis Baratheon instead of squandering it on fighting with Riverlords, Northmen and Valemen. Varys explained this in AGOT as:
“A boy with an army,” Varys said. “Yet only a boy, as you say. The king’s brothers are the ones giving Cersei sleepless nights... Lord Stannis in particular. His claim is the true one, he is known for his prowess as a battle commander, and he is utterly without mercy. There is no creature on earth half so terrifying as a truly just man. No one knows what Stannis has been doing on Dragonstone, but I will wager you that he’s gathered more swords than seashells. So here is Cersei’s nightmare: while her father and brother spend their power battling Starks and Tullys, Lord Stannis will land, proclaim himself king, and lop off her son’s curly blond head... and her own in the bargain, though I truly believe she cares more about the boy.”
Cersei knew she was better off with a cooperating Eddard Stark rather than a dead Eddard Stark as Varys explained:
Cersei is no fool. She knows a tame wolf is of more use than a dead one.
Cersei could of course not allow Eddard to resume his position as Warden of the North and Lord of Winterfell. She did not trust Eddard with power. But she knew that Eddard was a man of honor and always takes his vows and promises with immense seriousness. So She made this offer:
“I want you to serve the realm,” Varys said. “Tell the queen that you will confess your vile treason, command your son to lay down his sword, and proclaim Joffrey as the true heir. Offer to denounce Stannis and Renly as faithless usurpers. Our green-eyed lioness knows you are a man of honor. If you will give her the peace she needs and the time to deal with Stannis, and pledge to carry her secret to your grave, I believe she will allow you to take the black and live out the rest of your days on the Wall, with your brother and that baseborn son of yours.”
To break it down, Cersei had following demands:
- Eddard Stark will confess to treason
- Eddard Stark will command his son to lay down his Arms and go back to Winterfell or Join the Royal Armies against Baratheons.
- Eddard Stark will proclaim Joffrey to be true heir and denounce Baratheon brothers as vile rebels.
- Eddard Stark will swear to take Cersei's secret to his grave
If Eddard Stark gave her all this, Cersei had no reason to kill him and every reason to keep him alive because:
- It would free her from threats of Vale, Riverlands and North.
- It would give her time to deal with Stannis and Renly.
- It would keep her secret safe.
Cersei was aware that Eddard Stark would be able to visit his home as a Man of the Night's Watch but she knew he would never betray her if he swore a vow to keep her secret. So she had no fear on that count. Even if Eddard broke that vow, after his public confession of treason, Cersei could dismiss his allegations as self-serving lies of a self-confessed traitor. Also as a member of Night's Watch Eddard Stark will be prohibited by vows to play a role in Politics.
Impacts of Joffrey's little show
Unfortunately however, Cersei carefully planned schemes did not pay off. Her son sabotaged everything that she so meticulously built.
Eddard Stark was killed and what happened was exactly what she feared. Northmen were unlikely to accept any offer from Lannisters now. Riverlords were rallied by Robb Stark who crowned himself as King in the North and claimed lands of Riverlands and ancient Stark dominions. Given these developments, Her father was unable to move South and was forced to stay in Harrenhal to take on the Stark boy. Stannis Baratheon moved swiftly on King's Landing. Luck favored her when Highgarden joined Lord Tywin in relieving King's Landing and saving them from the Worst consequences. Tyrion managed to pacify Prince Doran with Myrcella's betrothal. Lady Arryn did not move as expected.
Cersei wanted to spare Ned's life to show a good face to the public. He was convicted of high treason by saying that Cersei's children are not Robert's, but Cersei said that that was a lie and had him punished. He should have been killed, but in order to be shown as a good Queen Regent, she said he would be sent to the Wall.
She would send a message to the realm that said "Despite the fact that he is a traitor, I spare his life and sent him to the Wall"; she would also show that she is helping the Wall by sending a proved warrior to serve. If she killed him, she would seem cruel -> just like people saw Joffrey. She didn't want that.
I think she believed that it was really hard for someone to believe the fact about her children.