Ha, Ha! Sorry guys! I just realized I tried to reference a variable in my script that wasn't there, and that was why it wasn't working. It was just a rookie screw-up, and I'm sorry I bothered you with it. Anyway this is how the code is supposed to look:
set-location "C:\Users\TMahoney1979\Desktop\Test"
foreach ($fn in get-childitem -name)
{
$splitname = @($fn -split " - ",2)
$newname = $splitname[1]
rename-item -path "C:\Users\TMahoney1979\Desktop\Test\$fn" -newname $newname -force
# Notice the variable $fn in the previous line, instead of $oldname, which was incorrect.
}