I've worked in academic laboratories, and I've been a software consultant. Nonetheless, the following is all speculation...

And my guess is that you'll find this to be a hard sell. I certainly wouldn't try it unless you can find other folks in this line of business who can tell you the ropes, or at least provide the existence proof that it works. If you can't find anyone else, that is a very bad sign.

The first problem is the packaging. It's easier to sell a PHP consultant than a generic "I'm a smart person who can learn your language of choice in a weekend and then hack for you" consultant. It's easier to sell a Drupal consultant than a generic PHP consultant. And it's easier to sell a Drupal consultant who is the world expert on, say, Drupal ecommerce or migrating data into Drupal than it is to sell a generic Drupal consultant.

So starting a consultancy is (surprise, surprise) a lot like starting any business: You need to find product-market fit. The good news, though, is that unlike the startup world you're not racing to find product-market fit: Your product by definition will not scale (if the customer could buy the solution in a $89 box from Google, they obviously wouldn't hire consultants) so you can just find other consultants and clone their product. If the market is at all healthy, those consultants will probably help you clone their product, and thank you for it: A bigger ocean floats all boats, and the others need a steady supply of new blood to help them pay the hotel bills for their industry conferences.

So if there's a market for, e.g., SAS consultants, learn SAS and call yourself a SAS consultant. Is there a well-known standard tool, or up-and-coming-standard tool, in your industry that everyone wants to use but nobody knows how to set up? That's a good candidate for a marketing hook.

I think you've identified your other, larger problems: A) Customers want in-house employees because learning the ins and outs of the customers' data and techniques is a long-term process, and B) everyone is super-secretive, because data is incredibly expensive stuff to acquire and leaking the data at the wrong time is like handing your competitors bundles of thousand-dollar bills. To that I would add problem C): Your target market is either (i) academics, who have relatively little money, relatively long timelines ("I'll request some grant money to pay your consulting fee; I'll let you know in a year if we get accepted"), and access to grad-student labor at below-market rates; or (ii) big companies, which tend to be able to afford in-house staffs, and will do so, because their secrecy concerns are even larger than usual.

There might be a category (iii): small-to-mid-size companies that can't afford an in-house statistician. Maybe you can find and target that market. But it might not be fun: By definition your market doesn't have much money, so they're going to try to hire you at below-market rates; they're going to try to get you to accept grand promises of future wealth in lieu of cash; they're going to nickel-and-dime you every step of the way, risking the quality of the work in the process; and of course there's the exciting possibility that they won't pay you at all. Make friends among the client's clerical staff, and have them keep an ear open for signs of potential bankruptcy. ;)

My suggestion is that you figure out what service the in-house statisticians inside well-funded companies would pay for, and then offer that. Don't try to be the rent-to-own in-house statistician for a company that has no statistician; Instead, offer a service that an in-house statistician would love to have but can't find the time or skill to do on her own. Of course, how does one best learn what in-house statisticians really want? By being one for a while. Take one of those jobs that are throwing themselves at your feet, hold it for a couple of years if you can, and tell yourself that it's market research for your future consulting firm.