LU64+ wii's are wii's that have a serial number that starts with LU64+ (USA), LEH250+ (EUROPE) or LAH113+. The + means that it also counts for numbers above, not that the serial continues.
NOTE: The serial number is not a surefire way of knowing if the console is affected by this issue. There are some consoles with LU64+ numbers but don't have this problem and a few consoles with earlier numbers that do. The only way to know for sure is to run the HackMii Installer to see what revision of boot2 is installed. Consoles with boot2v4 are affected by this issue.
Why are these Wii's so special? First, let's look at the Wii's boot process:
BOOT0 -> BOOT1 -> BOOT2 -> (Preloader) -> System menu
Like other consoles introduced earlier in 2008 LU64+ ones have an updated version of boot1 that effectively blocks BootMii installation at boot2. The more important difference, however, is the updated version of boot2 shipped with these consoles (boot2v4). This version of boot2 takes slightly longer to execute after being launched by boot1. Most older versions of IOS appear to be incompatible with boot2v4 due to the differences in timing between boot2v4 and the two earlier revisions.
Therefore any console with boot2v4 installed cannot safely be downgraded to any firmware below 3.4, nor will any IOS associated with an earlier firmware work once installed (for example, IOS36 v1042 from 3.1). Known exceptions to this rule are IOS16 v257 and IOS15 v257.
The obvious solution here is to avoid downgrading your System Menu or individual IOS's to pre-3.4 states except where proven safe by the community.
For more details, see bushing's series of posts on hackmii.com:
Timing is everything (the case of the “unsoftmoddable Wii”)