Considering that Stannis almost certainly knew and was at large, this was not a serious issue at this point. Cersei would let him live to extract concessions from him, and as it turned out, he would have made an extremely useful hostage when her brother and cousins fell to Robb. In wars between Houses like this, there is a kind of exemption for nobles, because they are too valuable to kill outright. You can get a lot of money in ransom from them or you can trade them to get your own people back and so on. Whatever "lies" Ned might go around telling about the people who beat him in the struggle for control of Robert's succession, alive he had far more use.

Anyone who wanted the throne was a threat to Joffrey's survival, and part of the deal was Ned taking the black. Once he did so, he would be bound to take no part in conflicts among the realm. Considering that the Wall is half the kingdom away from any border or place of contention, that statement of neutrality would be rather pointless if it was simply limited to refraining from sending troops to a fight. By taking no part, it would have to mean that Ned cannot testify against or endorse any claimant. And he certainly could not call his banners or lead the North against Joffrey & the Lannisters. From Cersei's perception, Catelyn and Robb were not a threat, as no one had ascertained Robb's martial prowess.

That's not a right call either. She's a noble of House Stark, and Stannis Baratheon is her rightful liege lord. If she has to die to allow the truth of a usurper's false claim to come to light, that's tough noogies. That sort of risk is the price you pay for getting to live in the only home in the North with internal heating, with getting fine clothes and lessons from a septa and lemon cakes. You get the best medical care, food, shelter and entertainment available in your time and place, and in exchange, you have to sometimes make other sacrifices, like having to marry unappealing individuals for the good of your family, or be endangered for political reasons.

Also, the threat to Sansa was largely BS. For the Lannisters to murder an 11-year old girl over some unknown intransigence on their part would blacken their name even worse than the incest long term, just as the Red Wedding did for the Freys. No one would ever trust the Lannisters with hostages in a peace treaty again, or other situations requiring a similar degree of trust. Quite simply, at that point, there was no publicly acceptable reason to kill Sansa. What is more, a hostage is your only bolt. The Freys get stuck on this dilemma with Edmure in aFfC when they threaten to hang him, but Brynden Tully tells them to go shove it. They are reluctant to carry out the threat, because then they'd have no hostage once he's dead. Likewise if Ned continued to hold out and Sansa was killed. Now, not only would he have no motivation for making peace, but the Starks would be even more likely to be intractable and fight to the knife. Note how exchange of hostages seems to be typical ending to a fight where both parties survive - the Greyjoys gave up Theon when they bent the knee, Robb asks for hostages in exchange for a peace treaty with the Lannisters, and Tyrion the same. Clearly, this is a desirable condition in order to prevent one side or another from going back on a deal, but who would ever trust the Lannisters again if they murdered a prepubescent child to get her father to surrender? Look at the horror among the Starks over the murder of hostages by Rickard Karstark, when they had already murdered Ned, and there was no sign of Arya - in other words, a lot more justification than existed to kill Sansa. Ned should have known all this, but Varys was smart enough to see that his weakness was his loved ones and his irrationality about harm to them, and given his sufferings and privations, was even less likely to be thinking clearly. So Varys offered him a way out (note that he made no direct threat - merely suggested a possibility of them bringing him her head) and Ned jumped at the excuse to cave in.