Why are there so many times Harry was told about him having his mother's eyes?

Question

I can't help but notice that this has been said to him a plethora of times.

How many times was Harry told about him having his mother's eyes?

Was there something important behind the frequent mention?

Answer 1

To start off, the mother's eyes were very important to J.K. Rowling. This is confirmed with this interview with JKR and Daniel Radcliffe:

Rowling: And I said, um, “The only really important thing is that his eyes look like his mother’s eyes. So if you’re casting Lily, there needs to be a resemblance, but they don’t absolutely have to be green.” (Deathly Hallows 2 DVD)

Also:

Q: Do you know what Harry's parents look like?
A: "Yes. I've even drawn a picture of how they look. Harry has his father and mother's good looks. But he has his mother's eyes and that's very important in a future book." (Loer, Stephanie. "All about Harry Potter from quidditch to the future of the Sorting Hat," The Boston Globe, October 18, 1999)

Of course, we know now that JKR was referring to Snape's story as told in DH.

In addition, the eyes seem to be a pretty big characterization point for JKR - she frequently mentions other characters' eyes, more often than other appearances.

• As an interesting side note, JKR mentioned in CoS that Dobby's eyes were green as well - and of course, he sacrificed his life to save Harry, as Lily did.

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TL:DR: Counts

Mentions of Harry's Green eyes:  20
Of them, comparisons to Lily's:  9 direct, and one implied.
Of them, told to Harry (may be indirectly): All 10
Of them, 2 or 3 are of extremely critical importance to the plot.

Count	Mentioned by	Mentioned to	Harry+Lily?	Snape's memory seen by Harry?	Cite?
1	Hagrid	Harry	1		HP1:CH4
1	Ollivander	Harry	0		HP1:CH4
2	Harry	Harry himself	2		HP1:CH12
2	Dwarf 2/14	Harry	0		HP2:CH12
1	Dumbledore	Harry	1		HP3:CH22
1	Mrs Weasley	Harry	0		HP4:CH10
1	Rita Skeeter	Harry + ALL	0		HP4:CH18
1	Elphias Doge	Harry + ALL	1		HP5:CH3
1	Narrator	Harry	1	Yes	HP5:CH29
2	Slughorn	Harry	2	Yes	HP6:CH4
1	Narrator	N/A + Harry	0, Implied		HP7:CH22
1	Narrator	Snape	1	Yes	HP7:CH23
1	Dumbledore	Snape	1	Yes	HP7:CH23
4	Narrator	N/A	0		HP1:CH2;+ HP2:CH1; HP3:CH1; HP4:CH2

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The most important: Snape and Dumbledore, in DH (CH22/CH23):

When the falsk was full to the brim, and Snape looked as though there was no blood left in him, his grip on Harry’s robes slackened.
“Look…at… me…” he whispered.
The green eyes found the black, but after a second, something in the depths of the dark pan seemed to vanish, leaving them fixed, blank, and empty.

So, before his death, Snape wanted to see Harry's Lily-like eyes.

And, of course, the most important - the Snape's memories from DH:

Lily’s bright green eyes were slits. Snape backtracked at once.

.

“I thought…you were going…to keep her…safe…”
“She and James put their faith in the wrong person,” said Dumbledore. “Rather like you, Severus. Weren’t you hoping that Lord Voldemort would spare her?”
Snape’s breathing was shallow.
“Her boy survives,” said Dumbledore.
With a tiny jerk of the head, Snape seemed to flick off an irksome fly.
“Her son lives. He has her eyes, precisely her eyes. You remember the shape and color of Lily Evans’s eyes, I am sure?”
“DON’T!” bellowed Snape. “Gone…dead…”
“Is this remorse, Severus?”
“I wish…I wish I were dead…”
“And what use would that be to anyone?” said Dumbledore coldly. “If you loved Lily Evans, if you truly loved her, then your way forward is clear.”
Snape seemed to peer through a haze of pain, and Dumbledore’s words appeared to take a long time to reach him.
“What – what do you mean?”
“You know how and why she died. Make sure it was not in vain. Help me protect Lily’s son.”

The eyes - the resemblance to the love of his life - was what drove Snape to help Dumbledore to the end.

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Mentions to Harry by other characters:

... 'Yeh look a lot like yer dad, but yeh've got yer mum's eyes.' (Hagrid, PS, CH4)

... 'Yes, yes. I thought I'd be seeing you soon. Harry Potter.' It wasn't a question. 'You have your mother's eyes. It seems only yesterday she was in here herself, buying her first wand. Ten and a quarter inches long, swishy, made of willow. Nice wand for charm work.' (Ollivander, PS, CH4)

"Here is your singing valentine: 'His eyes are as green as a fresh pickled toad,'..." (Singing Dwarf, CoS, CH12 - no comparison to mother)

This was partly because he didn't think he could stand Fred and George singing, "His eyes are as green as a fresh pickled toad" one more time (same place)

'An easy mistake to make,' said Dumbledore softly. 'I expect you're tired of hearing it, but you do look extraordinarily like James. Except for your eyes - you have your mother's eyes.' (Dumbledore, PoA, CH22)

... his dress robes didn't have any lace on them at all; in fact, they were more or less the same as his school ones, except that they were bottle green instead of black. 'I thought they'd bring out the colour of your eyes, dear,' said Mrs Weasley fondly. (Mrs Weasley, GoF, CH10 - no comparison to mother)

"Tears fill those startlingly green eyes as our conversation turns to the parents he can barely remember." (Rita Skeeter's article, GoF, CH18 - no comparison to mother)

'Oooh, he looks just like I thought he would,' said the witch who was holding her lit wand aloft. She looked the youngest there; she had a pale heart-shaped face, dark twinkling eyes, and short spiky hair that was a violent shade of violet. 'Wotcher, Harry!'
'Yeah, I see what you mean, Remus,' said a bald black wizard standing furthest back he had a deep, slow voice and wore a single gold hoop in his ear 'he looks exactly like James.'
'Except the eyes,' said a wheezy-voiced, silver-haired wizard at the back. 'Lily's eyes.' (OotP, CH3)

It was one of the girls from the lake edge. She had thick, dark red hair that fell to her shoulders, and startlingly green almond-shaped eyes. Harry's eyes. (Snape's memory as seen by Harry, OotP, CH29)

'You look very like your father.'
'Yeah, I've been told,' said Harry.
'Except for your eyes. You've got –'
'My mother's eyes, yeah.' Harry had heard it so often he found it a bit wearing. (Horace Slughorn, HBP, CH4)

And, what is undoubtedly second most important occurrence in the series:

'You're a good boy,' said Professor Slughorn, tears trickling down his fat cheeks into his walrus mustache. 'And you've got her eyes... just don't think too badly of me once you've seen it...' (Horace Slughorn, HBP, CH22)

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Mentions by Harry himself:

She was a very pretty woman. She had dark red hair and her eyes – her eyes are just like mine, Harry thought, edging a little closer to the glass. Bright green – exactly the same shape, but then he noticed that she was crying; smiling, but crying at the same time. (When first looking into the Mirror of Erised, PS, CH12)

... And slowly, Harry looked into the faces of the other people in the mirror and saw other pairs of green eyes like his, other noses like his (same place)

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Mentions by the narrator:

Harry had a thin face, knobbly knees, black hair and bright-green eyes. (PS, CH2 - no comparison to mother)

Harry, on the other hand, was small and skinny, with brilliant green eyes and jet-black hair that was always untidy. (CoS, CH1 - no comparison to mother)

Harry, though still rather small and skinny for his age, had grown a few inches over the last year. His jet-black hair, however, was just as it always had been: stubbornly untidy, whatever he did to it. The eyes behind his glasses were bright green (PoA, CH1 - no comparison to mother)

A skinny boy of fourteen looked back at him, his bright green eyes puzzled under his untidy black hair. (GoF, CH2 - no comparison to mother)