Macintosh

I have an old 1,1 Mac Pro running 10.6.8. Mostly the Eve client works fine, but I see this problem from time to time.

There are two things I've found that can cause/prevent this. The first is switching to other apps after hitting the Log In button on the Launcher. The problem seems to be that the launcher needs to start the wineserver (background app that runs cider) and then start a game client that the wineserver knows about, and then the client resets the screen resolution (the part where the screen goes black briefly). In order to do all that the launcher has to bring wineserver and the game client to the foreground. If you switch to another app that just happens to not want to give up foreground status, it breaks the game client spawn process.
tl;dr After you hit login, don't switch to any other apps while the Eve client is loading. Only safe to do so after the screen flashes black and then the character screen loads.

The second problem is with that wineserver background app. If wineserver starts up and doesn't properly attach itself to the game client, or if the client crashes in certain ways, the wineserver process does not shut down. So the next time you try to use the launcher, it tells the wineserver to attach to the new game client, but wineserver still thinks it's attached to the old client and so fails to do anything. No error message, just an icon appearing in the dock then disappearing.
The thing to do if the launcher repeatedly fails to spawn a game client properly is to open up the Activity Monitor software. Set it to "All Processes" at the top, click on the column header named PID (to sort by Process ID), and look for wineserver near the top of the list. If you have a wineserver open, but no Eve Online process open, use the Quit button to get rid of wineserver. This may possibly leave behind errors in system memory, however it generally solves the problem. The alternative is to restart your machine.