I'd like to see a quality open source Basecamp clone put up on Github.
Written in Rails of course, ironically. There certainly are enough Rails devs driving open source projects to make this happen.

That's the peril/blessing of embracing open source—given enough time, no matter how good you are, you will end up competing with free and/or a free clone of yourself!

Then let folks just arrange for their own hosting(Heroku could become perfect for this), or host it locally, and install & run it themselves. What if many, many more companies would use a Basecamp-like app if they could run it
inside their corporate firewalls?

Basecamp is not rocket science. Its core is a dozen pages give or take, and a bunch of standard CRUD. It does look polished, very well designed and thought out, and is highly usable. It's a showcase example of what a great web app should be. Yet almost all of these unconventional parts that seem like they would need heavy lifting configuration could be automated away.

Sure it's a bit of a dick move to make a white room clone, even though it happens all the time, and is almost the very definition of the majority of popular open source software. However, after several years, once some innovative software becomes mainstream, can it still claim exclusive right to the ephemeral methods of implementation? Once the knowledge of those methods, and the usage patterns are learned and spread throughout a community, the value of the knowledge plummets. Before it became obvious to everybody, sure, it would have been enormously difficult to create Basecamp from scratch. But after everyone has seen Basecamp and used it, it's funny that it now seems obvious that is how it should be done. It's like the software patent issue essentially.

I'd be curious to know the true cost of each Basecamp user in terms of hardware, network and storage cost. What if it's oh say $1.00 on average? Then it would be profitable for some company like Heroku who specializes only in streamlined, small Rails app hosting to butt in and sell hosting for a cheap price like $5. The software is free, the hardware is not, to steal a page from the Apple play book. That would bleed 37signals' profit margins. Isn't it fun to play race to the bottom pricing of software as it crystalizes into a commodity over a few years? Sure a few years ago, Basecamp would be difficult to make. But 2 years from now?
I don't know if the assumption holds that 2 years from now it will still be non-trivial to clone Basecamp in such a way we haven't even thought of nor seen on the web. Markets move quickly, especially in software. Everyone knows competition is a bitch, especially when the goal posts keep moving, and while the technology playing field itself undergoes rapid, constant geologic & volcanic upheaval.

The workflow in Basecamp itself is a great template for creating other kinds of web apps outside of project management and address books. Many other kinds of businesses have similar workflows that could be represented in a souped up Basecamp without too many modifications. An open source clone could also serve as a good boiler-plate starter app for developing more customized Basecamp-like apps. What if the value of that is more profitable than selling Basecamp itself?