Why is it when you (or maybe its just me) sometimes listen to white
noise i.e. the humming of the air conditioner, for example, you
perceive to hear music or sounds as well?  What is the term for this
illusion?  Is this due to me going to sleep mode?

Yes it is only you -- OK I lie.

Comes into the category of sensory deprivation.

On a personal basis, when quite young I used to ring bells up a dark
and gloomy perpendicular church tower to call the faithful to worship
(there is an irony there BTW). Only thing I could possibly hear for
the 15 minutes was the extremely loud bell, but I sure heard lots of
other things. Real creepy.

Also motor-biking, very high ambient noise but there are all these ?voices?.

In both cases the effect was / is more pronounced when dark,
reinforcing the sensory deprivation explanation. ?Drifting off? a wee
bit certainly enhances it too -- perhaps you are aware of dreams when
you are reasonably awake, or have attuned yourself to this skill. As
you have probably found, it?s fabulous listening to the music.

Best

Good comment.

Funny, I too used to ring church bells and had the same experience. 
What I can't figure out it the random sounds in the white noise that
is triggering the illusion or the sensory deprivation?

I see white noise as all this random sounds mixed together.  Is my
brain trying to make sense of it and creating a pattern ie music? 
Interesting. There has to be a term for this, it seems too common.

Weird that we both rang bells. Maybe it?s a bell ringer phenomenon :-)
?The BELLS, Esmeralda, the BELLS.?

Yes you?re right -- one would expect something so common having a
specific name. Couldn?t find one though. But there?s no name for the
back of the knee either. I think perhaps that the understanding is so
poor and non-definable that it just comes under the general term of
?hallucination?, which of course it is. Several times I have been so
tired that I have experienced severe visual hallucination but I think
the dividing line between this (plus other related phenomena) and
schizophrenia is that I was aware that what I was seeing wasn?t real.
I have a young relative who?s personal stereo is permanently glued to
his ears because he says it stops him hearing voices. Well, I hear
voices too but have never been diagnosed as schizophrenic, as has he.
Granted my (and your?) ?voices? are only under the previously stated
conditions but one can appreciate how close to the surface these
effects are.

Dunno if you have electronic experience but AGC (automatic gain
control) is employed in many applications. Lacking an input, the gain
ramps up and if unrestrained finally resolves the noise floor. This is
similar to the operation of our senses whereby a constant, non-varying
stimulus dulls the associated sense. If you stare at something for
long enough your vision of that object disappears. Same applies to the
sense of touch. Also a background noise such as a ticking clock, a
humming transformer or even a radio. So deprived of the stimulus (even
though it should actually still be there) the nervous system?s
analogue of AGC gets super sensitive and creates things from within.
Personally I tend to believe that we are always dreaming, even when
fully awake, though that?s far too big a subject to explore here.
However, therein may lie the source of these hallucinations.

But this still doesn?t resolve exactly why, normally only in
conditions of noise does one hallucinate sounds. Not exactly true
though because here in the county it is sometimes absolutely pitch
black and utterly silent. Even my cat can?t see. After a time, one
does begin to hear and see things. Hallucination, but nothing defined
and more like the noise floor becoming apparent. Some might argue that
the brain extracts details from white noise and reconstructs it in the
most rational manner which it can. I don?t subscribe to that though.
Somehow, I feel, the connection to ones inner brain workings becomes
more facilitated and dreams (which I mentioned) leak out -- much
perhaps as in the case of schizophrenia.

A few pages which you might like to explore, equating in some
instances the links I have made.

http://www.priory.com/halluc.htm

http://www.contac.org/contaclibrary/research32.htm

http://www.medhelp.org/forums/neuro/messages/30750a.html

Clinical psychology is still largely only in the state of data
collection and collation so I suspect there just is not yet an answer
to your fascinating query. I may be wrong of course, and would be
delighted if someone else could enlighten us. I guess though, for your
absolute answer, post again in five years time and I may have worked
it out -- or perhaps you will have.

Best

Our brains are constantly seeking patterns. If no pattern exists, one
may be fabricated.

I wish I could hear music in white noise. Ever since I was a child, I
have heard voices whispering in sounds such as radio static or running
water, and I'd rather not hear them. Unlike some folks, I don't think
these are the voices of the dead or of spiritual entities. I think
they are illusory patterns imposed upon chaos by my busy brain, which
insists upon trying to make sense of everything.