I have a softmod with a very old 20 GB hard drive (it was old even before it was installed in the Xbox). This drive sounds like it's on its way out because it clicks very loudly.

I understand the stock BIOS cannot map hard drive space larger than 137 GBs. But I have a 160 gig hard drive on hand... would it still work without the BIOS update? In other words, would the Xbox simply see it as a 137 gig drive? I'm not too concerned about losing that extra 23 gigs, since I doubt I'll even fill up the rest of the drive anyway (the current drive only has about 14 gigs of space occupied).

Modern softmods don't have that limitation. If you don't know which softmod you have, use HeXEn (Google Heimdall's Xbox Engineering Disc) burned to DVD-R at 4x or slower to upgrade your softmod to Krayzie 1.1.1 or Kingroach NDURE 3.1. You can then use HeXEn's lxhdm implementation to build the new disk, which in turn uses XBPartitioner 1.1 to format the new drive properly so the softmod can read it all.