As a book reader, I just wanna throw the following in (also because it was asked for):
We can't know from the books what Ned mumbled in the seconds before his death, because the chapter is a Arya chapter, everything is described from her perspective. And she's holding the feet of Baelor's statue (just like in the series) hundred, maybe two hundred meters away from Ned.
So no way she could have heard those words, even if they existed.
Sansa stood nearby, she also has chapters that look back to this moment, but I don't think it's ever mentioned she saw or heard anything from Ned. Also she was standing in the wrong geometry to have heard anything.
No, clues, if anything could come from Ned himself. During the chapters when he's sitting in the dungeon, yes he thinks a lot about his family. But a recurring theme, also in his dreams is Lyanna and her "Promise me, Ned!".
It seemed to me that he has way deeper feelings of 'unfinished business' towards Lyanna, which haunted him way more than anything to do with the family alive.
Based on this, I'd find it not surprising if someone would tell me his last words were "I've kept my promise."
It's a long time ago however since I've read those chapters so I'll admit this is opinion-based.
And of course D&D knew where Martin was going, yadda yadda, etc. and something we couldn't know from the books ended up in the series.