Harry's patronus is a stag. It pretty obviously is a direct reference to his father's animal form (His father's patronus was also a stag). Hermione's patronus is an otter, and Harry's mother's (and Snape's) is a doe. Does these imply anything about personalities? What are the factors that may lead to the form that a corporeal patronus will eventually take?
There's no precise canon answer, but some tangencial information that may have some bearing:
Q: Does the animal one turns into as an Animagi reflect your personality?
JKR: Very well deduced, Narri! I personally would like to think that I would transform into an otter, which is my favorite animal. Imagine how horrible it would be if I turned out to be a cockroach! (source: Rowling's America Online chat transcript, AOL.com, 19 October 2000)
While the quote is about Animagi form, we know that they 2 are somewhat related (most if not all known Animagi have the same Patronus as Animagus form - Prof. McGonagle, James Potter are 2 I remember off the top of my head).
Also, as has been discussed previously, JKR indicated that falling in love can affect one's Patronus. Whether you consider that part of "personality" is kind of subjective.
On the other hand, some Patronuses seem to have zero relationship to their caster, such as Ron's:
MA: What's Ron's Patronus?
JKR: Ron's Patronus? Have I never said that either? Oh no, that's shocking! [Laughter.] Ron's Patronus is a small dog, like a Jack Russell, and that's a really sentimental choice, because we've got a Jack Russell. He's insane.
(source: Anelli, Melissa and Emerson Spartz. "The Leaky Cauldron and Mugglenet interview Joanne Kathleen Rowling: Part Three," The Leaky Cauldron, 16 July 2005)
Quite clearly, Ron is NOT insane :)
Oh, as far as the relationship between otters and Jack Russell terriers, apparently the dogs are known for chasing otters? That's what I've read elsewhere.
It really does seem to me that everyone's Patronus represents them in some way.
Otters, like Hermione's, are clever (they use tools like rocks to break into their food and can handle puzzles) and inquisitive. Clearly that's pretty suitable for Hermione!
Luna's is a rabbit, which has a host of sybolism. In Japan it's associated with the moon (like her name); one source says that in Kerry, hares were thought to hold the souls of the villagers' grandmothers, and Luna could be said to be an old soul; they're a symbol of fertility and new beginnings, and the "new beginnings" part seems to fall very much in line with her personality somehow; and finally, rabbits are born with their eyes open so superstition says they hold powers over the evil eye, and of Luna it could be said that her eyes are more open than others', both to the creatures she believes in which nobody else does, and to the nature of the people and the world. She's pretty aware. So I'd say her Patronus is really fitting, as well.
Ginny's is/was a horse, and horses are thought of as spirited and free, which matches her personality. I can't find it now that I'm looking, but I read once a superstition that said, "because horses can find their way in the dark, people believed they could foresee danger or could guide souls through the underworld." If I tried I could probably spin together something about how she "found her way through the dark" time that she was under Riddle's diary's control to become the confident person she later was, and that she "became a guide" to Harry in his own dark times by becoming one of his closest friends and then his wife.
Lupin's is easy enough: Not only does his family seem to have some kind of affinity with wolves (assumed because of their names), but because of his condition and the way it's treated in the wizarding world, it is pretty clear that his werewolfishness is something which causes him some great consternation. He resents it, but he can't separate it from his personality as a whole; to him, the werewolf is who he ultimately is, and what he's a slave to. To his mind, it's not only a representation of himself, but also his entire history and maybe his fate.
I do not even know what is up with the Potter family. Harry's Patronus was a stag before he knew that his dad was an animagus, so it was independent of that information - but was James's Patronus a stag before or after he could turn into one? Was Lily's Patronus a doe before or only after she fell in love with James? The Potter family is just hemorrhaging deer, it's crazy. Deer are agile and difficult to capture, though, which fits with the family's association with Voldemort... and there's a lot of dignity to a deer in mythology, especially in its association with the forest, as there seems to be much dignity attributed to the Potters, as other wizards speak of them. But seriously, this group gives me some trouble.
In general, though, I think it can be said that there is a lot of evidence to support a person's Patronus being a reflection of some intrinsic part of their character or personality. It's that way with a large enough sampling of the characters with known Patronuses to feel like a pretty good bet.
The Patronus, represents that which is hidden, unknown but necessary within the personality.
Rowling's Pottermore writing about Patronuses, includes some quotes from Charms of Defence and Deterrence by Professor Catullus Spangle.
No reliable system for predicting the form of an individual’s Patronus has ever been found, although the great eighteenth-century researcher of Charms, Professor Catullus Spangle, set forth certain principles that are widely accepted as true.
The Patronus, asserted Spangle, represents that which is hidden, unknown but necessary within the personality.
‘For it is evident,’ he writes, in his masterwork ‘Charms of Defence and Deterrence‘… that a human confronted with inhuman evil, such as the Dementor, must draw upon resources he or she may never have needed, and the Patronus is the awakened secret self that lies dormant until needed, but which must now be brought to light...’
Here, says Spangle, is the explanation for the appearance of Patronuses in forms that their casters might not expect, for which they have never felt a particular affinity, or (in rare cases) even recognise.
I was discussing this with a friend lately and it might be that Hermione's otter Patronus is a reference to Ron. The Weasleys live in East Devon (hence supporting Chudleigh Cannons), and it's implied that it's in the Otter Valley since the nearest village is Ottery St Catchpole...
I've been lead to believe that a person's patronus is dependant on the person they love the most dearly.
- Ron's patronus is a Jack-Russell Terrier, which are known to chase otters. Coincidentally, Hermione's patronus is an otter. This reveals that Ron had had a crush on Hermione in The Order of the Phoenix, or even earlier on in their school lives.
- Snape's patronus was a doe, like Lily's, because he loved her so much and her death affected him enormously. Like Tonks, his patronus changed to reflect the person with whom he had a strong connection.
- Tonks's patronus became a wolf after she met Lupin.
- Harry's patronus was a stag like his father's, because of his resemblance to James, in appearance, personality, and interests (like Quidditch).
I think that, in Harry's case, it's because he loved his dad, even though he hadn't met him. In Tonks's and Snape's case, their love was unrequited: Remus in Tonks's case and Lily in Snape's. If, say, Snape started to fall for someone else, would his patronus change too?