I was named after my father, as was my son.  While he was alive, he
was Senior, I was Junior, and my son was III.  Now that my father has
passed away, proper etiquette says I should become Senior (or nothing
at all) and my son's name changes from III to Jr.  But what about all
the legal documents that have the old suffixes (birth certificate, SS
number, credit accounts, deeds)?  How would a name change affect them?
 Or should I just keep things as they are?

THis is just a free comment and no "answer" to your question, and of
course it is no legal or professional advice, as you can read on the
disclaimer below.

If the birth certificates, SS number and other legal documents show
you as "Jr." and your son as "III", I believe that that is the way you
remain.  These sites discuss the necessity for continuing to use  -
consistantly -  suffixes:

https://dklbweb.dekalbga.org/voter/pdf/applic.pdf
http://www.southernct.edu/faculty/banner/index.php?file=standards.html
http://www.dmvs.ga.gov/forms/pdf/motor/t-227.pdf

I guess that you could have your names changed to make you "Sr." and
your son "Jr.", but your dad most probably was just referred to as
"Sr.", since until you were born, there was no way of knowing that
there would be "Jr."
It would probably be a nuisance to change you names because then your
son would have your name while documents exist that are in your old,
"Jr." name.  Messy!

Keep Dad's/Granddad's remembrance alive by recognizing that he was "the senior".

Cheers, Myoarin

My dad is a Jr. and my brother III and my grandfather was a Sr., but
he died long before my brother was ever born.  People ask Dad all he
time why Brother is III if Grandpa is deceased and Dad tells them
because he is a Jr. - that's how is was born and that's how he'll
always be, and that makes Brother a III.  I think that it is OK to
remain a Jr. and your son a III if you want that.  In fact, I know
quite a few Sr./Jr./III and none of them changed anything after one
died.  If you wanted to change your name to Sr. you certainly can, but
I don't think that it is inappropriate for you not to.

Actually there is no etiquette rule or even traditional following that
gives a definitive answer.  Etiquette experts are pretty much split on
the "moving-up" dilema, as it is called.  I am a IV and both the JR
and SR of the family have died, and neither me or my father have
changed or suffixes.  I believe you should just keep it as is, it
would get confusing if you did change it.  Plus it adds authenticity
if you know how far back your name goes.

I believe you should take thomas224-ga advice.
Look at the history.
Henry the IIX,John Paul II and so on.
It is Histroy you should not try to rewrite it.