Here's another thought although I haven't checked. One thing that can cause problems is the command "get". In general when you run a command like "name of me" the command get is implied so you're really running "get name of me". The problem is that the implied "get" is not always the case. So sometimes you have to explicitly say "get". Whenever I have a problem like yours the first thing I try is to add "get" to the command... it's become habit because you just never know. Note that you can always use the word get and never have that issue. As such, try changing your command to "set my_name to (get name of me)". I'd be interested to know if that fixes your 10.5 problem. Also note that a name is already a string so there's no need to coerce the result to a string.
EDIT:
I looked through some of my older scripts. I used the following code to get the name. In my notes I have these comments...
-- this will get the name of the application or script without any file extension
-- it is done using the path because when a script is run from the script menu, and you write set myName to name of me, then the result is "applescript runner" instead of the actual name
-- also it assures that you're getting the name as it appears in the Finder because sometimes the system events process name is different than the Finder name
on getMyName()
set myPath to path to me as text
if myPath ends with ":" then
set n to -2
else
set n to -1
end if
set AppleScript's text item delimiters to ":"
set myName to text item n of myPath
if (myName contains ".") then
set AppleScript's text item delimiters to "."
set myName to text 1 thru text item -2 of myName
end if
set AppleScript's text item delimiters to ""
return myName
end getMyName