Curious: why? Can't you simply shut down the NAS^wmicroserver and replace the failing drive then? (assuming the downtime is not a problem, which is probably the case with most home NAS installations)
One fear I've heard from Sys Admin friends is that you'll have another drive failure when the machine tries to boot back up (and possible data loss if you're only running RAID5). As I understand it, there's a nonzero chance the drive is fine while it's still running, but will disappear or have trouble spinning back up after a reboot, so you want to rebuild your array before rebooting.
Is there any truth to the advice that this is mitigated by buying drives from different retailers and aiming for different batches and average failure time?
Having different batches reduces the risk of multiple devices failing at the same time. There's a higher chance of devices from the same batch failing simultaneously (due to the same material or fault during the manufacture) than from different batches, or even better, from different vendors.