Finding files in subdirectories from one location

Using ssh, is there a simple command to find files in subdirectories from, let's say, the home directory on a shared server?

Yes. You can use find command. For example:

find /home/user -iname "nastyfile"

That will search /home/user for any file (case insensitive) with the name "nastyfile".
For more options (which are plenty) read man find.

Thank you.