It came across to me as if Catelyn basically insisted that Eddard take the job because if he didn't it would make Robert not trust him any longer, which would in turn lead to problems down the road with the kingdom. Also, because of his relationship with Jon Arryn, who was something of a father figure to him. The only way to potentially find out about his death would be to take the job so he could investigate the matter. He seemed very reluctant to agree to this.