Try searching around wowwiki, there's usually a list of common names for each of the different races. Generally, just go for a water-themed name, that rolls off the tongue in a fluid (gettit?) manner.

I don't know of any name generators, but there are a few ways to go about RPing the concept of owning an Elemental.

The more literal interpretation is that your mage really has enslaved (or perhaps befriended) a real water elemental, either the Azerothian variety that would have formerly served Neptulon, or possibly the more primal and less stable Outland varieties. Most of these Elementals tend to follow the same naming conventions, with names that tend to be references to water or storms, usually ending in -ion or -is. There's a list of notable Water Elementals in the game here, which gives an idea of what they tend to be named : http://www.wowwiki.com/Water_elemental.

This idea actually has some interesting possibilities for RP attatched to it, especially if your mage happens to know a Shaman. Is the Elemental being forced to serve you (which a Shaman would likely frown upon), or is there some kind of a pact going there? If it's a pact, what is the Mage's end of the bargain that needs to be upheld? Was the Elemental a more friendly variety that a Shaman friend may have helped find or mediate a good relationship between?

Another explanation is that it isn't a "proper" Water Elemental at all, but rather a familiar created by the Mage themselves, in which case it's something of an extension of their own magic rather than a completely separate entity serving them. You have a lot more flexibility with naming and personality in this case, since it would be up to the Mage to name their creation and decide whether or not to give it a personality (which again is likely an extention of their own). There's a lot of potential for RP here as well. Does the Elemental's personality represent a specific aspect of the Mage's own identity? Is it maybe a portion of the Mage's personality that isn't very strong, created as an attempt to compensate for that deficiency (for instance, a shy and introverted Mage creating a friendly and outgoing elemental)? Do their personalities clash, or do they always get along? Is the Elemental made to be completely subservient, or has it been given the ability to think for itself, raising the possibility for disobedience and arguments?

tl;dr : There's a lot of possibilities for RPing a Water Elemental, and it's a really fun concept to explore. If you want it to have a "proper" Elemental name, then look at the named Elementals in-game to get an idea of the naming conventions they follow.

I named mine "Innuendo" cuz I couldn't pick just one.

I'm not sure if this strays too much from the names of most other Water Elementals, but I thought Spritz fit pretty well. He helps when I summon water, after all.

Ah noo, mine's called Spritz too!
My main mage (yes I have more than one) has a water elemental called Basil. I always called him that and it's great that it's actually the name above his head now.

Bonds, Hydrogen Bonds. (was that already taken?)