Im starting a website and I am going to use a company to do data market research for me to determine the interest in my idea. How many respondents would be significant to satisfy potential partners/advertisers of my site?
Typically, a sample size of 2,000 responses would be considered statistically significant. You can get as low as 50 - 200, but you need to employ some statistical weighting methods do adjust for the lower number of absolute responses ( a graduate student with a statistical background from a local University, could easily help you with that - perhaps as part of an internship?).
I agree with Robin's point, though, that you definitely should be watching your trend data, and (depending on what and how you are trying to measure) be willing to adjust based on the results / the direct feedback from your target market.
It all depends on what you are doing and how you are doing your research.
A research paper, prepared by a worlds class market research company, that shows in a statistically significant way the there are billions of people that should be interested in your site is worthless (because the hard part is getting them to the site and convincing them to buy, click an ad or do whatever you need to get money, the fact that they are out there does not make you money).
On the other hand if you make a one page website, mention it somewhere and almost immediately get 2 - 5 people e-mailing you and asking to buy your non-existent product this is a very strong signal the product can be a success (because it shows that the customers are out there, they are looking for something like your product, they managed to find you and they are willing to pay).
What you are looking for is not a statistically significant result, at least not in this stage (getting statistically significant result before you have any traffic is either impossible or very expensive) what you are looking for is a signal (not proof) that customers exist, you can reach them and they are willing to give you money (or click an ad, or whatever) without all 3 you don’t have a business and measuring this without doing some real marketing is impossible.
You need to achieve statistical significance. Realistically this means >1% percent of your prospective market ...
Practically though, it depends on what your doing. If your site is intended to have 1 million uniques a day then the answer differs from someone who is going to have a maximum of 5000 dedicated users in total (if each user is paying $100-$500 per month you still make good money).
Without more details I would be guessing 50 answres is a weak trend, 100 users starting to show a trend is a start, 1000 showing a trend is good, more is better.
Really don't stop tracking ... every day/week you should use these are your own guide. Keep publishing them as your advertisers will change over time and it will keep your parnters in the loop and still thinking of you.