Why did Ned Stark keep this secret about Jon Snow from Catelyn Stark?
That Jon Snow is not a bastard, but the son of his sister, Lyanna Stark, and Rhaegar Targaryen.
After all these years Catelyn still hated Jon Snow. She can't accept him as a son. It didn't mean she hadn't tried but she was just not able to do it (She had conversation with Talisa Stark about how she is not able to accept Jon Snow as her own son)
Ned Stark knew that Catelyn hated Jon Snow.
Didn't Ned think that, above all of others, Catelyn is the one who deserved to know the secret about Jon Snow after all these years of hatred and pain she had on this matter?
I think this would have eased things between Jon Snow and Catelyn Stark. He knows he is a bastard, but he has good relationships with all of Ned Stark's sons and daughters, yet, because Catelyn Stark hated him, he always feels that he is an outsider.
Or was Ned also planning to tell the truth to Catelyn about Jon Snow's mother just like he promised to Jon Snow before he went to King's Landing as Hand of the King?