I've been having issues with this also. Its very critical that our business have remote access to our terminals. Lion has pretty much stopped us dead in our tracks. Rebooting does not help. Hopefully Logmein will provide an update soon.
I have updated to the latest version, manually, as they describe in the blog. It doesn't fix the black screen issue. I can log into my White i-Mac (non-Lion), and connect to my MacBook Pro via screen sharing (that way I know my Mac Book Pro is not asleep), but LogmeIn does not work directly to the MacBook Pro (Lion).
I get the same black screen when remoting into a 2010 iMac 27" upgraded to Mac OS X Lion 10.7 from Ignition on iPhone 4 and iPad 2, and from Free LogMeIn running on a new Mac Mini Lion. Oddly enough, ipad, iPhone and the iMac can all remote into the Mac Mini Lion perfectly. The Mac Mini is running HDMI to a Samsung 52" 1080p HD TV, running Display at 1080p. So, the only Ignition or browser based black screen problems I have are when trying to remote to a desktop Mac, iMac. But when remoting to the Mac Mini using an external TV display all is well
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".
Why oh, why do software vendors act surprised when a new OS comes out? It's been in beta for months, being Mac developers, logmein couldn't have tested this before Lion was released? It's not like this was top-secret, everyone knew it was coming in July & Lion betas were freely available to developers.