According to Wizarding World, Hermione's patronus (the otter) represents her affinity for these animals, as well as elements of her character.
Hermione is bookish and the otter is an especially child-book-friendly animal.
Her Patronus, the otter, is a clever and dexterous animal; some have
even been known to juggle small rocks. Otters have also featured in
many classics of English literature such as Tarka the Otter, Ring of
Bright Water and The Wind in the Willows. Trust Hermione’s Patronus to
be in so many books.
But the otter also reflected an aspect of Hermione we weren’t used to
seeing. According to the eighteenth-century Charms researcher
Professor Catullus Spangle, the Patronus ‘represents that which is
hidden, unknown but necessary within the personality’. To face
something as evil as a Dementor, a person ‘must draw upon resources he
or she may never have needed, and the Patronus is the awakened secret
self that lies dormant until needed.’
Hermione’s carefree side was usually buried beneath endless stacks of
homework, yet her Patronus animal frolicked around playfully without a
care in the world.
Hermione’s Patronus, a shining silver otter, was gambolling around her. ‘They are sort of nice, aren’t they?’ she said, looking at it fondly.
Harry Potter and the Order of the Phoenix
By comparison, Ron's patronus is more of a character trait alone.
Terriers are small dogs that seriously overestimate their own size and strength. Utterly fearless, they’ll bark furiously at anything they see as a threat, no matter if the odds are against them. Ron Weasley often showed this trait, by protecting Harry from an escaped convict while standing on a broken leg and defending Hermione’s honour with a broken wand, which resulted in an unfortunate bout of slug-vomiting.
Stubborn and wilful as the Jack Russell can be (remember Ron’s epic sulk during Goblet of Fire?), a dog Patronus is a surefire sign of a loyal friend. Take Sirius Black for example, a character most in tune with his canine side. Sirius was extremely loyal to his friends, his cause and to his godson Harry, whom he fought with until the end. Even if Ron went off in the odd huff, he always came back with a new-found ferocity. Man’s best friend indeed.
Luna's hare is evidently related to her madness and connection with the moon, something shared in Eastern philosophy with the moon rabbit
The phrases ‘mad as a March hare’ might well have described Luna
Lovegood, who had an interesting relationship with her Patronus. The
hare has a strong association with the moon, with the ‘moon gazing
hare’ being a symbol in ancient pagan beliefs. How interesting that
Luna’s name is of Latin origin, meaning moon.
Though some Patronuses reveal a hidden or repressed quality in the
caster, Luna always seemed to be fully in tune with her emotions. Her
nickname, ‘Loony’, comes from the word ‘lunatic’ or ‘lunacy’ – a
condition once believed to be caused by the full moon. Her Patronus
shows that she’s proud to be herself, no matter what anyone else says.
There's also the more mundane fact that her house is set alone on top of a hill in the English countryside. Rabbits would be found literally everywhere.
