There are many different types of web startups, so this is difficult to answer reasonably.
But, for example, a visionary will be very passionate about how they see changing the world, but then reality will eventually settle in and they will either adjust to reality or continue to fight in what they believe. For example, Second Life had a vision, but the reality of the types of enterprises were not what they expected, but they have adapted, and it seems modified their vision.
Then there are those that just want to monetize an idea that someone else envisioned, as they expand the idea, for example, what it seems happened with Facebook.
So, which type of business describes yours? Perhaps you are somewhere in between these three. :)
I think that a single founder is going to be in trouble. Successful startups almost always require a team, even at the very beginning. I gave a brief talk about this recently; you can see the slides about half-way down http://www.caycon.com/downloads.php under "Presentation: Building a High-Performance Startup Team."
In the presentation, I distilled the essential characteristics down to six C-letter words:
Standard qualities would be things like persistence, focus and passion. However there are more subtle things that come as a founder matures and gains experience. These are thing like knowing when to stop running around like a headless chicken and take a break. Hard work is obviously important but so is relaxation, that is when the best solutions to problems and great ideas come to you. I believe it is important to get that balance right early to avoid unnecessary mistakes and burnout.
I definitely agree with Daniel. It is important also to have balance and know when to stp and see if it is time for a different approach.
Coming from my own experience, along with studying about founders from startups big and small, I see the most successful founders having these 2 qualities:
If you have 1, then the dedication and passion will fall into place, even on the days you want to give up. This doesn't not mean the idea isn't tweakable as you progress. Sometimes the idea is a means to an end (ie Flickr). But overall, look at Google. They believed in their idea in spite of the 100's of me too search engines already available.
Quality #2 is obvious. You need to be the best at what you do to add the most value to your startup. This can be in sales, technology, or domain knowledge of a particular niche. Also, if you have this then you have the confidence of your team, employees, and investors.