I got a Best Time, but it didn't save it or ask me for my name
There are several possible reasons for this:
You might not have been using a standard difficulty level. You can only save high scores for the standard difficulty levels, Easy, Medium, and Hard. If you change the size of the board or number of mines, you can't save your high score. See Changing the size of the board.
You might have made more mistakes than the previous person. If mistakes are allowed, someone who takes longer but makes less mistakes is better than someone who takes a shorter amount of time but makes lots of mistakes. This is to prevent someone who sets the number of allowed mistakes really really high and then clicks on each square really really fast from getting a better score than someone who actually played the game.
You can only get a best time if you actually won. Although the fastest game would be to lose after 0 seconds, it's not impressive: anyone can do it.