カイリKairi (カイリ). Kai means sea, and Kairi's full name is homophonous with the word for nautical mile. Personally, I think it suits her very well, and not only because I like the name to begin with. But what's so significant about it?
Kairi is so named due to the trend of elemental thematic naming within Kingdom Hearts; Aqua, Terra and Ventus share the same naming scheme. For her trio, Kairi represents water, whilst Riku represents earth, and Sora represents air. But Kairi is given water. This really suits her role in the trio, in my opinion. Think of it a little like this; Sora is the sky, and Riku is the land. The sea is between land and sky, as Kairi is between Sora and Riku. She can bring them together, and also cause rivalries between them. She can break them apart but she can bring them back together in one stroke. In the darkness - the realm of darkness - there is the sea. And that is how she brings her two boys back to her. Yes, they return through the Door to Light, but it's from the sea where the bottle appears, where they land in Destiny Islands. Kairi has a lot to do with the sea.
Alongside this, is the fact that the sea is similar to the sky and to the land but the sky is not similar to the land. She's somewhat at the centre of things - as the sea she is a free spirit. She can go wherever she pleases, and her legacy expands wide, befriending and meeting many people; this is similar to the sky. She is thoughtful, intuitive, and somewhat grounded; this is similar to the land. She perhaps gets on with both Sora and Riku equally as a result, but in different ways, with different aspects of her personality (at least, that's how I read it).
Either way, this is quite speculative, so what about other name meanings? Depending on which kanji is used Kairi can mean multiple things, two of those being estrangement/separation, and dissociation. Dissociation is perhaps a little farfetched, but estrangement and separation definitely suit her. She was torn away from her home world, she was torn away again from her new one, she lost her friends, she waited for her friends, so many times... it's Kairi, in a nutshell. She waits. She waits. She waits.
Then, like the sea, she comes crashing down against those who stand against her.