Why did Eddard "Ned" Stark never mention or clarify that
Jon "Snow" is not his bastard son but his sister's, Lyanna Stark, son?
This answer is partly from the books, but as most of the Rhaegar and Lyanna theory evidence is from the books I'll assume that's ok.
Consider that if Lyanna and Rhaegar are the parents of Jon Snow, then Jon Snow would be a Targaryen. As such he would almost certainly be killed by Robert, given that Robert wants all surviving Targaryens dead (like with his assassination order of Daenerys). Jon would infuriate Robert in particular because Robert believes Lyanna was raped by Rhaegar, so he would stand as a reminder / insult to Robert.
Remember that Eddard had just had a falling out with Robert over the murder of Aegon and Rhaenys, so Eddard would likely expect a violent action from Robert if Jon Snow was Rhaegar's son.
IF what you say is true... It's a popular belief but, as I say in my comment, has yet to be confirmed by the author/in the next books.
The question is tagged game-of-throne, which is the series tag. In the series, I don't remember any clear evidence that Jon is not Eddard's son.
When Eddard sees Jon for the last time (when he leaves for King's Landing whereas Jon leaves for the Wall) he says something like "I will tell you everything you want to know about your mother the next time we meet". So I guess he just didn't plan to die before telling the truth.
If you want a book-based answer, the most likely to be is that Lyanna probably asked Eddard not to tell the truth: she made him make swear something. We do not know what, but probably to claim Jon as his son. See Warwick's answer for details about the reasons for that.
There appear to be two main reasons why this was the case.
Ned is an honourable man and he promised Lyanna he would not tell anyone.
Lyanna Stark: You have to protect him. Promise me, Ned.
Game of Thrones, Season 7 Episode 7, "The Dragon and the Wolf"
When we found out about Jon's true parents we also found out another little secret that Jon was actually a legitimate child of Rhaegar Targaryn and Lyanna Stark. As such he is actually a Targaryen.
Bran Stark: Jon isn't really my father's son. He's the son of Rhaegar Targaryen and my aunt, Lyanna Stark.
Bran Stark: He's never been a bastard. He's the heir to the Iron Throne. He needs to know. Game of Thrones, Season 7 Episode 7, "The Dragon and the Wolf"
This is why Lyanna asked Ned to protect the child, if Robert had found out he would have killed him.
Lyanna Stark: Please. Listen to me, Ned. If Robert finds out, he’ll kill him, you know he will. You have to protect him. Promise me, Ned. Promise me. Promise me, Ned. Promise me.
Game of Thrones, Season 6 Episode 10, "The Winds of Winter"
This is for the same reason he wanted Dany dead.
Robert Baratheon: The whore is pregnant.
Eddard Stark: You're speaking of murdering a child.
Robert Baratheon: I warned you this would happen. Back in the North, I warned you, but you didn't care to hear. Well, hear it now. I want 'em dead, mother and child both. And that fool, Viserys as well. Is that plain enough for you? I want them both dead.
Game of Thrones, Season 1 Episode 5, "The Wolf and the Lion"