For my second (yes, second) RRoD on my 1st generation 360, not only did they replace the faulty optical drive, but I'm pretty sure they replaced my board with an "Opus" board. Can't be sure without cracking it open, which I haven't, but 1) I used a power supply from a new 2009 360 (with HDMI) when at a friend's house, and it fit -- the newer power supplies are keyed so not to fit in the old systems, and 2) I had to re-authenticate all my DLC, even though my serial number stayed the same. Was a PITA, too, since the transfer didn't initially take and I had to wait for high-level Live support to reauthorize my box. Regardless, my 360 is now practically silent when the optical drive is spun down. YMMV, but that was my experience.
Regardless, you should get it fixed anyway. Even if you replace it, at least you'll have an extra unit to sell or trade.
Random Rage. Because your mother said so, that's why.