Is there a word for: the practice of responding to a "question" with
an answer that really doesn't fit the question?

Non sequitur?

http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=non+sequitur

Nonresponsive.

There was a wonderful question (no, it was wonderful comments) about
this as a trait of the fairer sex.
Can't remember that "non sequitur" was mentioned, but it really
wouldn't have been apppropriate there, since what the ladies gave as
an answer not only followed the idea of the question but move on ahead
to answer what could be the next question or what they surmised was
the real question.  (Q & A all within the context of that question and
not as ga Q & A), here:
" The female psyche and pre-emptive answers to unasked questions"
(I don't know how to call up a "thread".)

I find that this is because, in my view, women see things in a more
connected manner, whereas men see things as disconnected facts. This
shows in the male tendancy to bandy trivia about (like this) and in
their logical ability, where functions are performed one after the
other, and in the female ability for languages, where words are
connected in more than one way. This causes women to think more about
the subjects related to the question rather than the question itself,
as men would.

This also gives them the advantage over men that, when the question is
misunderstood, they still answer the original (I presume, through
talking so broadly that the question's answer is encompassed). Men do
not have such a tendency in reaction to a misunderstood question.

This probably leads to my mother's belief that you should never ask
for directions from a man. Either that or because the men my mother
has met have insisted on relating their personal memory of the route
rather than vague directions.

Pinkfreud and Denco have provided a couple of good options.  In
politics, where it's commonly used because the speaker controls the
platform, it's called a "dodge":

"To avoid fulfilling or answering completely: duck, evade, hedge,
sidestep, skirt. See seek/avoid."

Answers.com
http://www.answers.com/dodge

Best regards, 

Omnivorous-GA

Evasive (evade) is a great one by Omnivorous-GA.  Good show!