Is there any reason to pronouce Aaron Copland's composition "Rodeo" as
Ro-DAY-o instead of Ro-Dee-o?

Technically, either "ro-DAY-o" or "RO-dee-o" is correct. The former is
the pronunciation in Spanish, which is the word's language of origin;
the latter is more popular on the rodeo circuit.

Dictionary.com: "rodeo"
http://dictionary.reference.com/search?q=rodeo

However, since Copland himself used the pronunciation "RO-dee-o" for
his composition, I prefer that pronunciation.

"Shortly after this transitional period, the artistic populism of the
thirties begins to affect Copland's artistic direction. He wants his
work to reach people who don't normally go in for Horrid Modern Music.
He writes for popular venues: Outdoor Overture for high-school
orchestra, the 'school opera' The Second Hurricane, and music for
theater, ballet, and films. His collaborators, also affected by 30s
populism, choose well-known American mythic subjects, and Copland
responds (with the help of Virgil Thomson) by simplifying his musical
materials and incorporating folk influences. This results in his most
popular works: the ballets Billy the Kid, Rodeo (incidentally,
pronounced ROH-dio by the composer.)"

Classical Net
http://www.classical.net/music/comp.lst/copland.html

"Over the next two decades, Copland would continue to fuse folk and
other forms of popular music with his own classical compositions. In
1938, 'Billy the Kid' became an American ballet classic, a
romanticized account of the western legend. Copland's 'Lincoln
Portrait' created the musical atmosphere in which to present the
enduring words of Abraham Lincoln. 'Rodeo' (pronounced by Copland
'ROE-dee-oh') was created in collaboration with Agnes de Mille,
choreographer of the Ballet Russe de Monte Carlo, the organization
that commissioned the work."

Classical Music Detroit
http://www.classicalmusicdetroit.com/copland.htm

"The first acquaintance that I came to have with truly American music
was through the music of Aaron Copland. When I was a 'cellist in the
high school orchestra, we played 'Hoedown' from the Rodeo suite
(which, incidentally, I am told Copland pronounced as we do in Texas,
ro'-dee-oh.)"

Homepage of Dr. Gerald McDaniel, North Central Texas College
http://www.nortexinfo.net/McDaniel/american_music.htm

"Met Copland once when I was a student.  I was gratified to hear that
he pronounced the name of his ballet 'Rodeo' as RO-dee-yo, rather than
the (to my ears) affected ro-DAY-oh that most classical announcers
use."

Post from alt.appalachian newsgroup
http://groups.google.com/groups?selm=9EAF3BD5C7A3E5FC.A4D99D7CCF5494A7.10E36E8393513056%40lp.airnews.net&oe=UTF-8&output=gplain

I've found several online endorsements of "ro-DAY-o," but none of them
give any particular reason for overruling Copland's preferred
pronunciation:

"In the middle of the 1920's Copland had developed an interest in
jazz. He wanted to write music that was clearly American. He also
wanted to write music that concert audiences would love even after
finding out that he was a living composer. To that end, he borrowed
hymns from New England and folk music of North and South America. He
used the melodies of those songs as themes in his music. Perhaps his
most famous compositions are the ballets; Billy the Kid,
Rodeo,(pronounced ro-day-o) and Appalachian Spring."

Lancaster Symphony
http://www.lancastersymphony.org/pdfs/AaronCopland_Fanfare.pdf

"Buckaroo Holiday comes from Copland's famous ballet, Rodeo
(pronounced Ro-day-oh), where the lives of cowboys and cowgirls mix
with the music of the open prairie."

Pittsburgh Symphony
http://www.pittsburghsymphony.org/education/supplements/suppmaterials/Old%20West%20Notes.pdf

NOTE: The two files linked above are in .pdf format. You need to have
Adobe Acrobat reader software installed on your computer in order to
view .pdf files. If you do not already have this software, a free
download is available here:
 
Adobe
http://www.adobe.com/products/acrobat/readstep2.html 

All things considered, I think we do greater honor to this remarkable
work by using Aaron Copland's own chosen pronunciation: RO-dee-o.

My Google search strategy:

Google Web Search: "aaron copland" + "rodeo" + "pronounced"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22aaron+copland%22+rodeo+pronounced

I hope this information is useful. If anything I've said is unclear,
or if a link does not function, please request clarification; I'll
gladly offer further assistance before you rate my answer.

Best wishes,
pinkfreud

Exactly what I wanted and couldn't find myself (mostly because I was
searching for "pronunciation" instead of "prodounced.")