Hardware requirements for Linux server to run R & RStudio
There is no ideal configuration, for R or in general - product selection is always a difficult task and many factors are at play. I think that the solution is rather simple - get the best computer that your budget allows.

Having said that, since you want to focus on R development and one of R's pressing issues is its critical dependence on the amount of available physical memory (RAM), I would suggest favoring more RAM to other parameters. The second most important parameter, in my opinion, would be number of cores (or processors - see details below), due to your potential multiprocessing focus. Finally, the two next most important criteria I'd pay attention to would be compatibility with Linux and system/manufacturer's quality.

As far as the storage goes, I suggest considering solid state drives (SSD), if you'd rather prefer to have a bit more more speed than more space (however, if your work will involve intensive disk operations, you might want to investigate the issue of SSD reliability or consult with people, knowledgeable in this matter). However, I think that for R-focused work, disk operations are much less critical than memory ones, as I've mentioned above.

When choosing a specific Linux distribution, I suggest using a well-supported one, such as Debian or, even better, Ubuntu (if you care more about support, choose their LTS version). I'd rather not buy parts and assemble custom box, but some people would definitely prefer that route - for that you really need to know hardware well, but potential compatibility could still be an issue. The next paragraph provides some examples for both commercial-off-the-shelf (COTS) and custom solutions.

Should you be interested in the custom system route, this discussion might be worth reading, as it contains some interesting pricing numbers (just to get an idea of potential savings) and also sheds some light on multiprocessor vs. multi-core alternatives (obviously, the context is different, but nevertheless could be useful). As I said, I would go the COTS route, mainly due to reliability and compatibility issues. In terms of single-processor multi-core systems, your budget is more than enough. However, when we go to multiprocessor workstations (I'm not even talking about servers), even two-processor configurations can go over your budget easily. Some, not far away, such as HP Z820 Workstation. It starts from 2439 USD, but in minimal configuration. When you upgrade it to match your desired specs (if it's even possible), I'm sure that we'll be talking about 5K USD price range (extrapolating from the series' higher-level models). What I like about HP Z820, though, is the fact that this system is Ubuntu certified. Considering system compatibility and assuming your desire to run Ubuntu, the best way to approach your problem is to go through Ubuntu-certified hardware lists and shortlist systems that you like. Just for the sake of completeness, take a look at this interesting multiprocessor system, which in compatible configuration might cost less than from HP or other major vendors. However, it's multimedia-oriented as well as it's reliability and compatibility are unknown, not to mention that it's way over your specified budget.

In terms of R and R-focused software, I highly recommend you to use RStudio Server instead of RStudio, as that will provide you with an opportunity to be able to work from any Internet-enabled location (provided you computer will be running, obviously). Another advice that I have is to keep an eye on alternative R distributions. I'm not talking about commercial expensive ones, but about emerging open source projects, such as pqR: http://www.pqr-project.org. Will update as needed. I hope this is helpful.