How to debug a Python module run with python -m from the command line?

You can't do it now, because -m terminates option list

python -h
...
-m mod : run library module as a script (terminates option list)
...

That means it's mod's job to interpret the rest of the arguments list and this behavior fully depends on how mod is designed internally and whether it support another -m

Lets check out what's happening inside pdb of python 2.x. Actually, nothing intereseting, it only expects a script name to be supplied:

   if not sys.argv[1:] or sys.argv[1] in ("--help", "-h"):
        print "usage: pdb.py scriptfile [arg] ..."
        sys.exit(2)

    mainpyfile =  sys.argv[1]     # Get script filename
    if not os.path.exists(mainpyfile):
        print 'Error:', mainpyfile, 'does not exist'
        sys.exit(1)

    del sys.argv[0]         # Hide "pdb.py" from argument list

    # Replace pdb's dir with script's dir in front of module search path.
    sys.path[0] = os.path.dirname(mainpyfile)

    # Note on saving/restoring sys.argv: it's a good idea when sys.argv was
    # modified by the script being debugged. It's a bad idea when it was
    # changed by the user from the command line. There is a "restart" command
    # which allows explicit specification of command line arguments.
    pdb = Pdb()
    while True:
        try:
            pdb._runscript(mainpyfile)

Same for the currently released versions of python 3.x

Good news

The pull request that allows to do what you're asking has been merged 5 days ago. What a mysterious coincidence! Here's the code

So just wait a bit for the upcoming python 3.x versions to have this issue resolved )

Python 3.7 adds that feature

From the docs, it looks that your command:

python -m pdb -m my_module

will start working on Python 3.7:

New in version 3.7: pdb.py now accepts a -m option that execute modules similar to the way python3 -m does. As with a script, the debugger will pause execution just before the first line of the module.