Combining both: you can use custom error pages on Apache and make a tailored-to-your-site error message for the 503 error. I think the phrasing you used yourself is very clear:
"the system is experiencing higher than expected usage and the system timed out trying to log on the user" - You could adapt it a bit, but in essence this is clear and exactly what you want to communicate.
As for the 'be funny'-part, hmm, doesn't really go well with me. I would keep it to the point and not much else.
As for the look and feel: I would use the most basic elements (colorscheme) to make sure users know they are on the right site, and it makes it a bit less rough than either the big bold black letters on white background, or the internal IE messages with red exclamation marks... there is no need to panic after all. They just need to wait a bit. (Just put it all in plain html. Setting a few colors and a 3 line text doesn't put that much stress on the server.)
If this is an ongoing problem at every start of the day, you might of course look into either more beef on the server end, or try pre-caching common tasks to lighten the load.