Same issue here, Black Screen on Lion LogMeIn Version 1940 installed.
But i've found a solution.
The full story: i've installed the newest logmeinversion on an completly new Mac with Lion (factory installed). Logmein worked fine. after a reboot and reconnect with logmein i've got the OS X Login Page, i logged in an recieved the blackscreen. a reboot did not help (same thing happned "loginscreen->blackscreen). so i logged in again (blackscreen) and now opened an remote ssh session and shut down logmein and restarted it again via console. an now it works fine. but i have to do this again after a reboot. so i conclude the problem comes up when you need to login to your mac (on the mac itself, not logmein) at this point logmein seems to lose the "displayconnection".