TL;DR If your system clocks are not set to the same time, the Homegroup wizard will tell you that the password you are entering is wrong. Fixing the system time on one of my laptops was all it took for the Homegroup wizard to stop rejecting the correct password.
Of all the unhelpful messages Windows has ever rendered, this one takes the cake. It is downright misleading and seems like purposeful obfuscation. Homegroup is generally a super-convenient way to share resources across machines on the same network, but it's maddening when it fails out of the blue.
I have a Homegroup set up across three machines--Windows XP Pro x64, Windows 7 Ultimate x64, and Windows 8 Pro x64. The Windows 8 machine has spontaneously experienced two separate problems that cause its network shares to fail:
(1) Affiliation switches from home-network type WORKGROUP to business-network type DOMAIN. Fairly easy to correct but requires reboot.
(2) The only real trouble that Core2 Duo-era hardware seems to have with Windows 8 is imperfect BIOS reaction to sleep/hibernate/wake signals; I suspect a mismatch in this area can cause the system clock to drift. If your machines are not set to the same time (my Windows 8 machine was off by 35 hours) the Homegroup wizard will tell you that you are entering the wrong password.
Hand-setting your systems to approximately the same time should be good enough if you are unable sync them to the same internet time server.