I'm still not completely sure I understand what you want. You're copying multiple directories into one destination directory, but need to rename the directories in the process? You can't put a * in the destination like that, this isn't like DOS: The shell splits * for you before cp is even run, so you end up inputting stuff into cp that doesn't make sense if you * for things that don't exist yet.

I think you're going to have to split it into seperate cp calls.

Code:

for DIR in /home/*
do
        # Don't copy exp into exp/exp-timestamp!  It'd copy the copied copy copies!
        [ "$DIR" == "/home/exp" ] && continue

        NAME=`basename "$DIR"`
        cp -p "/home/${NAME}" "/home/exp/`date`-${NAME}"
done