Sirs, 
(Plz give this to "hummer-ga". They researched this b4 and have quick answer.)
Your answer can be either quick yes - you can and go for it...or no,
don't spend anymore of your time to pursue.
I am the US born son of an Englishman. My father was born (1903) of
English missionaries. We have documents that show he was born a UK
citizen and was one for part of his life, til he came to USA (1927).
At some point (this maybe the critical)...he became a naturalized US
citizen. I am not sure yet, if it was before I was born (1952) or
after.
A friend of mine just just got his UK passport (his mother was
English). He said to me that I may have possibility to qualify. It
would really help me, because I plan to live in the EU long term
(self-employed and also working for a US based company). Do I have any
chance to qualify for a UK passport and pursue this with the British 
Consulate in Los Angeles????????????

Hi  blueblueblue,

Thank you for referring your question to me. The quick answer is: "Yes
- you can go for it". However, although the answer may look quick,
it's not really because I had to do some research before coming to my
conclusion. Therefore, here are the results of my research. You'll
find what you need at the British Home Office website:

Chapter 2: Automatic acquisition - persons born before 1 January 1983 
http://www.ind.homeoffice.gov.uk/ind/en/home/laws___policy/policy_instructions/nis/chapter_2.html?

Follow the flow charts to determine your British status -

1) Determine if you were a CUKC before 1 January 1983:

Annex C:
FLOW-CHART TO DETERMINE WHETHER A PERSON BORN ON OR AFTER 1 JANUARY
1949 WAS A CITIZEN OF THE UNITED KINGDOM AND COLONIES ON 31 DECEMBER
1982 (where appropriate, "he" and "his" may be read as "she" and
"her")
YES -> NO -> NO -> YES -> NO -> WAS A CUKC
http://www.ind.homeoffice.gov.uk/ind/en/home/laws___policy/policy_instructions/nis/chapter_2.Maincontent.0004.file.tmp/CH2_ANNEXC.pdf

Answer: YES

2) Determine if you had the Right of Abode on 31 December 1982:

Annex D:
FLOW-CHART TO DETERMINE WHETHER A CITIZEN OF THE UNITED KINGDOM AND
COLONIES HAD THE RIGHT OF ABODE IN THE UNITED KINGDOM ON 31 DECEMBER 1982
NO -> YES -> YES -> YOU HAD THE RIGHT OF ABODE
http://www.ind.homeoffice.gov.uk/ind/en/home/laws___policy/policy_instructions/nis/chapter_2.Maincontent.0005.file.tmp/CH2_ANNEXD.pdf

Answer: YES

2) Determine if you became a British Citizen on 1 January 1983:

Annex E:
FLOW-CHART TO DETERMINE WHETHER A PERSON BECAME A BRITISH CITIZEN ON 1
JANUARY 1983
YES -> YES -> NO -> YOU BECAME A BRITISH CITIZEN ON 1 JANUARY 1983
http://www.ind.homeoffice.gov.uk/ind/en/home/laws___policy/policy_instructions/nis/chapter_2.Maincontent.0006.file.tmp/CH2_ANNEXE.pdf

Answer: YES

So there you have it. Print out the three flow charts, marked with a
highlighter pen and take them, along with your father's birth
certificate and documents to prove your identity to the consulate.
That will be enough to get you off to good start in your pursuit at
the British Consulate in Los Angeles. There's no guarantee that all
will go as you wish but I believe there's enough evidence to make it
worth your while to try. By the way, both Britain and the U.S.
recognize dual citizenship, but be sure to have a U.S. passport before
pursuing this and use it when leaving and entering the U.S.

Regards,
hummer

I vaguely remember the 'Two Grandparent Rule'

- talk to the consulate