Python: Fix missing import with a monkey patch
old_invoke is trying to reference mymodule's getopt, which doesn't exist. You need:
>>> import mymodule
>>> old_invoke = mymodule.MyClass.invoke
>>> def new_invoke(self, *k, **kw):
...   print("New invoke")
...   import getopt
...   
...   # here
...   mymodule.getopt = getopt
...   
...   old_invoke(self, *k, **kw)
...
>>> mymodule.MyClass.invoke = new_invoke

But, realistically, you should just have an import getopt in mymodule:
# mymodule.py
import getopt
...

Then your function is simply:
>>> import mymodule
>>> old_invoke = mymodule.MyClass.invoke
>>> def new_invoke(self, *k, **kw):
...   print("New invoke")
...   old_invoke(self, *k, **kw)
...
>>> mymodule.MyClass.invoke = new_invoke

Addendum
As another note, using import in a function isn't generally recommended. Unless you are only calling this function once (and even then, why?), every time you call the function you are attempting to load a module, which at best will always do a check against sys.modules when you probably don't have to.
Have the import getopt at the top of the script:
import getopt
import mymodule

mymodule.getopt = getopt

Which is where you'd probably expect this to be anyways

I am trying to monkey patch a missing import. The old_invoke() still does not get the import.
In case it is relevant, MyClass is a gdb.Command.
(gdb) pi
>>> import mymodule
>>> old_invoke = mymodule.MyClass.invoke
>>> def new_invoke(self, *k, **kw):
...   print("New invoke")
...   import getopt
...   old_invoke(self, *k, **kw)
...
>>> mymodule.MyClass.invoke = new_invoke
>>>
(gdb) my_gdb_command
New invoke
Python Exception <class 'NameError'> name 'getopt' is not defined:
Error occurred in Python: name 'getopt' is not defined

Also, in case it is relevant, the initial files and sourcing looks something like this:
mymodule.py:
import gdb
class MyClass(gdb.Command):
   ...

   def invoke(self, arg, from_tty):
      options, remainder = getopt.getopt(args, 'p:s:t:o:')
      ...

MyClass()

myothermodule.py:
import mymodule
...

Sourcing the above
(gdb) source myothermodule.py

