How can I do branding research online cheaply and quickly?

I'm just starting my business, I have an idea for a game that I think will be very popular. Trouble is, I can't decide on a good name for it that fits and is memorable. So I want to do some market research. I'd rather do so cheaply as I don't have much starting funds for the business. (In fact one reason I want to find a good name for the game is so I can start a kickstarter project for it!)

I'd rather find a way to do it for free, but can budget up to about $500 for the research if need be.

What methods/services do you recommend to achieve this quickly and inexpensively? Is this possible or am I better off just picking any old name and going with it?

There's no substitue for talking to real people.
Depending on where you are in the world, $500 could buy you about ±150 cups of coffee.
Pick some names, make some mockups, head out on the streets and coffee shops.

Apart from testing ideas on real people as Nick notes, you should read Positioning by Jack Trout and Al Ries. It will give you some simple tips and advice that will help you think about the name.

Hints: avoid acronyms since they are hard to remember. Avoid coined names like Xerox, Coca Cola or Mercedes since they don't actually say what the product or service is about. Try to find a name that states the major benefit you offer, that's generic and descriptive and that's easy to remember like Head & Shoulders, Intensive Care, Close Up, People magazine or BusinessWeek magazine.

When thinking about your name and positioning statement or tagline, always think in the way that people in your target audience think and use their vocabulary not yours. Don't try to coin a new term or concept or force something unfamiliar on your audience.

Avoid jargon like the plague and don't try to make up new words or categories. Leverage terms and concepts that are already being used to your advantage. One brilliant example of this was 7ups position from some time ago as "The Uncola."

Good luck!