Is there a way to use paid advertising to test a domain name?

Yes.

Use the "Link Text" in your facebook ads to A/B test various domain names.

To do this, simply launch a few identical ads, and only change the link text to say the variation of the domain name you are considering.

The ad with the highest CTR (not conversion rate or other KPI) will be the one that sparks the most interest in your target audience.

Note: one domain may have a high relevance to one audience, but low relevance with another. So this tactic should only be used if you have a very specific demo you are targeting.

The importance of domains names has diminished significantly and will likely continue to do so. Buy whatever is available with the fewest words, that is the easiest to spell. Or put another way, has less likelihood of being misspelled.

An easy and free way to test the popularity of specific keywords is to use google.com/trends. You can enter up to 5 keywords at a time and compare search interest in them over time. Then you could select a domain name based on the most popular keyword available as a domain name.

I am not sure what you mean by testing a domain name... I am assuming you want to see if one brand name works better than another... considering the cost of buying a domain name is less than $10 a year, I would just buy the ones you are interested in and try promoting both with adwords...

In general it is a violation of google's TOS if you dont have added value content on your site... so I wouldn't advertise it on google if you are just testing a domain name... I would make sure to have actual content and a functioning website...

I would recommend using Google Consumer Surveys to do a poll of two different names, but make sure to register both domain options or someone might steal them from under your nose... http://www.google.com/insights/consumersurveys/home?gclid=CKG_8Kfsv7oCFUfZQgodhS8Azw

Ordinarily, I try to avoid explicitly contradicting other experts on Clarity.fm. However, some of the comments above are preposterously incorrect. Tom Williams and David Melamed are probably very knowledgeable in their chosen fields. However, based on their remarks here, it's quite clear that they're comparatively inexperienced with domains.

"The importance of domains names has diminished significantly and will likely continue to do so." That's a bold prediction, but I see no reason for believing it. Has social media obliterated the importance of domains? OK. Then why did the founder of Facebook pay $75,000 for the domain name Internet.org last August? Sales of that size happen almost weekly. Or has Google condemned domains to worthlessness? OK. Then why is Google itself now competing for the privilege of running a fleet of new domain extensions such as .WEB?

And the other problematic statement: " ... considering the cost of buying a domain name is less than $10 a year ... " Which domain is that? Here is a list of last week's reported sales:

http://dnjournal.com/domainsales.htm

To those 2 experts I'm disagreeing with, it's nothing personal. However, the statements above are grossly mistaken; and it's important to point that out.

In answer to the original question: Yes, there are a few ways to use paid advertising to test a domain. A/B testing in this area is actually an underutilized but important ingredient in SEM. You should test the domains using whichever PPC channel you're planning to advertise in -- whether that's AdWords or something else.

While I would advise such an approach for extending your SEM dollar, I probably wouldn't advocate paid advertising as a test to choose a domain in the first place. Unless the intended domain were a 6 or 7-figure purchase, it would simply be too expensive to run a test like this long enough to gather statistically significant data.