I am looking for a children's book my mom read as a child.  She was
born in 1948.  She said the book was about a girl who lived in an
island/tropical environment (Hawaii or polynesian).  She had an old
coconut doll she played with all the time, inside and outdoors but
grew tired of it because it was ugly.  She wanted a new doll and got
one for her birthday, however the new doll could not be taken outside
because it would get dirty with sand.  She then discovered that the
best doll was her old coconut doll.

I am confident that the book you're looking for is "Ukelele and Her
New Doll," by Clara Louise Grant. This was one of my favorite books
when I was a little girl (like your mom, I was born in 1948).

From a forum for collectors of Little Golden Books:

"I would like to find a golden book that my mother had as a child,
here's what I know about it. A story about a little girl that lived on
an island. She had a doll made of either corn husks or coconuts.
Sailors came to the island and gave her a brand new doll which she
loved, she tossed the old doll aside. She eventually went back to the
old doll realizing new is not always better. My grandmother is
determined it was a Little Golden Book - time frame: My mother was
born in 1947 so I am guessing around 1950 - 1955...
 
I believe this story is 'Ukelele and her new doll'. It was published
in 1951. It is number 102. It was illustrated by Campbell Grant and
written by Clara Louise Grant."

GoldenBook.com: Little Golden Book Collectors' Forum
http://goldenbook.com/forum/viewtopic.php?forum=2&showtopic=415

Here's a photo of the book's cover art:

Ukelele and Her New Doll (cover)
http://www.mortalwombat.com/Special/ukelele.jpg

If you are interested in finding a used copy of "Ukelele and Her New
Doll," this book search will give you several options:

AddAll Rare, Unsed and Out of Print Book Search
http://used.addall.com/SuperRare/RefineRare.fcgi?id=061120154429397364

Here are two eBay auctions of "Ukelele and Her New Doll":

eBay: Ukelele and Her New Doll
http://cgi.ebay.com/LITTLE-GOLDEN-BOOK-UKELELE-AND-HER-NEW-DOLL-1951_W0QQitemZ230053530986QQihZ013QQcategoryZ279QQssPageNameZWDVWQQrdZ1QQcmdZViewItem

eBay: Ukelele and Her New Doll
http://cgi.ebay.com/RARE-1949-Edition-Ukelele-and-Her-New-Doll_W0QQitemZ110057916534QQihZ001QQcategoryZ279QQssPageNameZWDVWQQrdZ1QQcmdZViewItem

My Google search strategy:

Google Web Search: ukelele doll "golden book"
://www.google.com/search?hl=en&q=ukelele+doll+%22golden+book%22

I hope this is exactly what you need. If this is not the correct book,
or if a link doesn't work for you, please request clarification; I'll
be glad to offer further assistance before you rate my answer.

Best regards,
pinkfreud

Very fast and effective!  Great job!

This will be a great x-mas gift for my mom!  I can't wait to see the
look on her face!