In the first season of Game of Thrones, at the last moments of Episode 9, we see Ned Stark on his knees looking down. In the last few moments we see the sword being swung, his lips mumble something quietly, and the image cuts away.

Does it happen in the books, is it known what he was mumbling there?

Just A Prayer

Most recently, Sean Bean has revealed that he was "just saying a prayer".

[as transposed from the Huffington Post article]

Well, in an interview with HuffPost about his new show “The Oath,” Sean Bean set things straight.

Ned was ... “just saying a prayer.”

“It appears that way, doesn’t it?” Bean said, alluding to the fact that, yes, it does appear as though the doomed man is praying. “I couldn’t be too specific, because I don’t know if religion [like that] was around in those days, whatever they were. I just thought, ‘What would you do if this were really gonna happen?’ You probably would pray. You probably would murmur some words and you’d keep it quiet. You’d keep it to yourself.”

“It’s quite subtle in that many people wouldn’t pick it up,” he added. “It was an interesting thing to do for me at that point. There’s not much you can do really, you’ve got your head on a block. That’s about the only thing you can do is murmur.”
_{-Here's What Ned Stark Whispered During His 'Game Of Thrones' Death, Huffington Post, 2018-03-13, By Bill Bradley.}

It speaks to how first of all, it was Sean Bean himself who decided to go with this interpretation of the scene and that it wasn't actually planned. Therefore, it's not something that's going to have any deeper meaning.

It appears to be one of three competing theories but as far as I can tell there's no evidence to back any of them up apart from speculation.

1. Protect my family: Apparently this is what it said in some other languages either in their subtitles or the dubbed voices.

In the spanish translate it say "proteged a mi familia" which can translate to "protect my family" you can see it here (no subs)

I dont know if the translate is correct but I thought it could be helpfull.
_Nasluc

He says "protect my family".
We got the subtitles here in Brazil, and it says "Proteja minha família". The translation is exactly "Protect my family".
_fussomoro

2. I kept my promise: Though a lot of people seem to be wanting this to be true more than it is as it was referring to the "theory" R+L=J. Some "lip readers" on reddit have said he definitely doesn't say that though.

Im a lip reader and made an account solely to post on this. There's no possible way to get "I kept my" or "Lyanna" from his lip movements. I kind of understand how people got 'promise'. Close but wrong. Its actually 'protect' "Please protect Jon" clear as anything.
_KilTaco

3. Some prayer to the Old Gods

In my opinion he is probably saying a prayer to the Old Gods that includes asking them to protect his family.

There are of course some other theories such as "Please protect Jon", "I'm sorry Jon", "Forgive me", etc. But none of them really seem to have any value to them.

Another theory is "Valar Morghulis" but this makes no sense for a character like Ned, unless of course he was an Assassin's Creed like character and was going to reveal to his kids they were Faceless Men when they came of age. The only other way this would make sense is if someone took Ned's place, there are theories this is true but considering everything that has happened since then I highly doubt it.

Another theory is put forth in a reddit thread but it appears to be a joke as I can't track down a source.

In an interview with Sean Bean he told the interviewer he said:

"I'll survive the next role"
_lawlietskyy

As discussed in this Reddit thread, some lip readers have analyzed the scene pretty closely, and there are two main competing theories:

I kept my promise

Some posters add "Lyanna" after that. Ned made lots of promises, if he mentions Lyanna then he would probably be referring to the scene with her in the tower.

Protect my family

There are similarities here: "kept" vs "protect", "family" vs "promise", "family" vs "Lyanna". Some posters in non-English speaking countries reported having subtitles that translated to "protect my family". This could be a kind of prayer. His family is certainly in quite a bit of danger, as shown in later seasons.

Sean Bean just "said something to himself", a prayer he "sometimes does".

As seen in this interview with WIRED:

[Sean reading the question on a board] What does Ned Stark whisper before he dies?

It was just something to myself, obviously. I think it was a song, might of been... Gangnam Style, yeah that was it [in a melody] Gangnam Style [laughs].

No it wasn't. It was a little prayer that I sometimes do myself and I sometimes say. Because I wanted it to be something, I didn't want to just be mouthing something [mumbles gibberish] cause that don't mean anything and I think it changes things. So I thought I would do something that actually meant something to me. It was a plea for help in some way. So it did actually make sense to me and it was right to me and it was true.

_{Sean Bean Answers the Web's Most Searched Questions | WIRED Published on Apr 19, 2019}

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.