I'm wondering if there is a Greek or Arabic equivalent to the name
"Kimberly?"  (We're expecting a daughter, I'd like to name her after
my wife, my wife does not want a direct namesake, so I'm looking for a
less direct version of her name that comes from the heritage of one of
our families.)

According to this page, on the Diaper Dandies
naming service website, Kimberly is a name
which, in English, means "ruler":
http://www.diaperdandies.com/serv03.htm
(Use the "Find" function in your browser to
save you from scrolling).

According to this page of female Greek names,
Madora, or Medora also means "ruler":
http://www.20000-names.com/female_greek_names.htm


Searches done, via Google:

"greek names" meaning
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22greek+names%22+meaning

Kimberly Greek
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=Kimberly+Greek


If you need further assistance before rating this answer,
please feel free to post a request for clarification.

sublime1-ga

If you wish to find a modern greek name, one actually used in Greece
then the best is Despina, that means Ruler (in the Female form) and is
the equivelent of Madona in Latin. She will have a name day every 15th
of August that is a public greek holiday.

I knew a girl named Despina and she shortened to Depi. My sister's
middle naem is Despina to!

Despina is the Equiv. for Madona in Latin but for clarification they
are not translate the same.

In religion the Catholic will call Holy Mary, Madona, our Lady or My
Lady.
In Greek Orthodox She is called Maria Despina, Mary the ruler (female)
from the male Despotis, Ruler.

They are not translated the same but are used the same context.

Leon