Have you tried looking at the Utah State Historical Society Burials Database at http://history.utah.gov/apps/burials/execute/searchburials
? Be creative with the spelling of your surname. Try thinking of every way it has ever been misspelled, etc.
Also the Nephi Times News www.nephitimesnews.com
has donated copies of back issues to the Nephi Public library: see http://www.nephitimesnews.com/0508/050708/front.htm
. The Nephi Library's webpage is http://www.nephi.utah.gov/library/index.html
If you are near BYU, it lists the Times News in it's microfilmed newspaper collection http://www.lib.byu.edu/resource/newspapers/juab.htm
You will also want to look in Provo's Daily Herald http://www.heraldextra.com/
and The Salt Lake Tribune http://www.sltrib.com/
. Many residents of Nephi have their obituaries published there as well as in the local paper.
There is an effort to digitize newspapers and place them on the internet: http://dream.lib.utah.edu/digital/unews/
I don't know if anything that recent has been digitized yet.