Now you most definitely should read all of TDK's tutorials but the quick answer is Name$ needs to be set to GLOBAL for it to work inside functions.
The major problem with your code (the thing that probably caused TDK to say "you simply don't write programs like that.") is that normal code (anything not in a function) is placed at the top, then an "end" so hitting a function doesn't cause an error, then any functions at the end of the code. Functions cannot be in the middle as you have done. The "wait key" at the end will never, ever be seen.
Another problem is that even though you know whats going on (being the programmer) you need to inform anybody else running it to know what you're wanting. Add a "What is your name? " in the INPUT so people won't be scratching their heads looking at a blank screen.
There's no need to wait 2 seconds after it saves... unless the goal is to annoy people.
There's no need to SYNC since there is no SYNC ON. The default setting will update the screen anytime there is a change. There is also no need for multiple WAIT KEYs. When a function is called it does it's thing and always goes back to the next line after the function call. You can put one WAIT KEY under SaveGame() to achieve the same results as multiple WAIT KEYs in an IF statement.
But don't worry about the WAIT KEYs in the function because the way the code is written there's really no need to do a check in the function to see if the file was written properly. An unproper save would mean your hard drive failed or some other major problem that has nothing to do with code. Just cut out that check and rest assured that it will work no matter what.
` Make the string useable inside any functions
global Name$
INPUT "What is your name? ",Name$
SaveGame()
wait key
` Always put an end before the first function
end
FUNCTION SaveGame()
IF FILE EXIST ("SaveGame.dat") = 1 THEN DELETE FILE "SaveGame.dat"
OPEN TO WRITE 1, "SaveGame.dat"
WRITE STRING 1, Name$
CLOSE FILE 1
CENTER TEXT 320, 160, "Save Succesfull! Press any key to continue."
ENDFUNCTION