Run the vacuum cleaner - or any other big white noise generator. Often calms cranky babies. Plus, their crying isn't so piercing so you can put up with it longer, which calms the cranky parents too! Also, is the sleeplessness possibly a side-effect of the treatment for the eczema? Discuss with pediatrician?

Also, that allergy possibility seems a good place to look.

And driving around is only a good idea if the driver isn't as exhausted as the baby! My sister used to drive around on the highway with the windows down, and her twins would have to close their eyes from the wind - which would make them sleep, tricky tricky!