Whether he's Robert or Rhaegar's son, why did Ned keep it from Cat? He seemed to trust her wholly.
Pretty simple. He promised somebody he wouldn't tell anybody.
True that.
Seems pretty reasonable to request that he could at least tell his wife, given the affect it would have on his relationship and that Cat was nearly as trustworthy as Ned.
ETA: Doubly so if the mother is Cat's sister.
Best way to keep a secret is to keep it to yourself. Cat's sister?
Lyanna is Neds sister.
Regardless, it just seems over the top to ask of someone already doing you a huge favor.
Lyanna: Cool, so you totally take the blame for this. And remember, you can't tell anyone.
Ned: Right, I'll do you a huge favor and take the fall. But when you say anyone, you mean anyone other than Cat, right? I mean, obviously I have to tell my wife.
Lyanna: No, you can't tell
anyone.
Ned: Dude, she's like the 2nd most trustworthy person in Westeros behind me. And I could get divorced over this, or at best have a wife who's pissed off at me for all eternity.
Lyanna: Well too damned bad, you should have thought of that before I ****ed Robert/Rhaegar and didn't wrap it up.
Ned: ...
Besides, I'd think the kid would be in more danger with Cat not knowing than if she knew. Thinking she was Ned's love-child, she pretty clearly resented him, and wasn't it pretty common in those times/that world for mother's to make their husband's bastards mysteriously "disappear"?