If I had to spec it out, I'd make the following recommendations:
- Get at least the recommended amount of RAM for your OS (not the minimum); 4GB is great for Windows 7, and I'd recommend 8GB for Windows 8 (but 4GB is just fine).
- Get at least a dual core CPU running at least 2.0GHz per core. This gives 2.0GHz potentially dedicated to your browser. More cores won't help unless you also use the Force.com IDE/do other intensive tasks.
- Get at least Chrome 31, Firefox 26, or Internet Explorer 10. This is by far the most important factor. The newer browsers are more efficient by several magnitudes of speed. Also, salesforce.com is shortly discontinuing support for IE 7, so you may as well upgrade.
Note that this isn't an expensive upgrade. I bought my new Windows 8 dual core laptop with 4GB of RAM for about $250, and my new Windows 8 dual core desktop for $350. The Developer Console runs beautifully on these computers. Both machines sport a Windows Experience Index of about 4.9. My work computer has a 3.4 and still gets the job done in Internet Explorer and Chrome.
Just upgrading your browser will give you a measurably increased experience. If your company (usually IT) is resistant, tell them that IE 7 is losing global support (hint: IE 7 doesn't appear on the top 12 browsers at statcounter.com). Also, make sure you're on the latest and greatest Chrome, but keep in mind that Chrome is currently benchmarking slower than IE 10, by quite a bit.
Developers traditionally have needed more resources than the average computer user (except for, of course, gamers), and despite the cloud-based nature of force.com, the tools are still quite down to earth, which means you must have a decent amount of hardware and updated software.
Edit: Supported Browsers specifically states:
Internet Explorer 7 and 8 aren’t supported for the Developer Console