Is there any way I can help a non-relative to become a US citizen?  I
recently discovered that a friend of mine is actually an illegal alien
and I would like to sponsor her or help her in some way to get a green
card and become a citizen.  It looks like the only way in is through a
relative or work visa (or student visa).  Is that right?

.. Is that right?

No. Anyone can sponsor. However you need to differentiate between 
'sponsoring' and preferences.

Being sponsor creates a guarantee that immigrant will not end up on welfare:
http://answers.google.com/answers/threadview?id=432919

It helps, but it does not create a preference.
 
Preferences are based on family relationship,
need for certain type of workers, etc

Preferences  are described here:
http://uscis.gov/graphics/services/imm_visas.htm

There is also
 Green Card Lottery 
http://www.nationalvisaregistry.com/visabull/visabullsept.htm

All possibilities are  listed here;
http://ted.behlendorf.com/who.htm

Entering illegally used to be an obstacle to immigration 
(breaking the law) and people on visitors visa had to leave first
and apply for immigration visa from the outside.
Then came different 'amnesties' 
http://www.washtimes.com/national/20041110-123424-5467r.htm

and it just became too complex. Best thing you can do for you friend
is to arrange (pay for) a meeting with an immigration lawyer.


There is a free informal discussion of INS issues on the Internet
http://groups.google.com/group/misc.immigration.usa/about

but, like this service, they are not a replacement for legal advice.

Getting married is a frequent avenue for legalizing immigration status,
but it would have to be real. Rent this movie:
http://www.imdb.com/title/tt0099699/
that is the second best thing, you can do, and more inspirational and funny.

Good luck.

Hedgie