What should I do if my advisor urges me to do a project that I have no interest or faith in? About a year ago when I was a first-year graduate student, my advisor wanted me to follow up on a project left behind by an honors student. I agreed because, at that time, I thought the topic was important and the findings were rather interesting. However, over the past year, I was busy with other projects and my qualifying exam, so this project was put on hold for almost a year. This summer, my advisor urges me to get it done as soon as possible. 

I'd be more than happy to, had I not realized the original project is not methodologically sound. My RAs who re-coded the original data share the same feeling: The experimenter made various mistakes and was overly flexible, yet all data points entered into final analyses. I don't want to spend months chasing after effects that don't exist. Moreover, due to the complex study design, I can foresee what a nightmare it can be to build computational models in the future.

I suggested radical changes that make the experiment more rigorous and subsequent modeling efforts more tractable, but my advisor refused my proposal several times, arguing that making any changes will only delay the starting time. I feel stuck: On the one hand, I don't want to pursue this project as it is since it's most definitely going to fail; on the other, I don't have the courage (or "moral capitals") to defy my advisor because I have put off this project for so long. To make matters worse, I only have one advisor, so if our relationship deteriorates, I have no one else to turn to.

Any advice on what I should do is much appreciated!
Sometimes it is necessary to "bite the bullet" and demonstrate that the method doesn't work. But you need to provide clear and irrefutable evidence in a way that can actually convince people, not just a vague claim.

A classic example of the failure to do this is the Challenger explosion: all of the information needed to delay the launch was known, but because the engineers who had the data didn't present it in a clear manner–just a raw output of data—their supervisors overruled them. If they had come in with plots showing failure of the O-rings as a function of launch conditions, anyone could have recognized the danger of the situation.

So if your advisor is insisting, you probably need to demonstrate why his approach  is infeasible or he will continue to insist upon it.