There are many myths and doubts about this title. Do we have evidence about the original intention of Beethoven naming his symphony after the so called fate motif?
The 5th symphony was first performed in 1808, and Morse code was invented in 1837, so there is no connection between them. Beethoven did not tell his publisher to name the symphony, though he did tell his publisher to name the 6th, which was first performed in the same concert as the 5th.
The name appears to have come from Beethoven's secretary and biographer Anton Schindler. The name Fate for symphony and motif appeared at a similar time, apparently from Schindler's report that Beethoven had described the theme as "Fate knocking at the door". Schindler might have been telling the truth, but there is considerable doubt.
I agree with Peter that Schindler, Beethoven's secretary, has told this story about the fate that's knocking at the door.
But we have enough reason to doubt about it:
even though Bernstein refers to this myth in his educational video: