I once asked a group of 12-year-olds what they would do if they were going to hide the spelling of someone's name in a composition. "I certainly wouldn't put it at the beginning," declared one wise young man.
The principle sticking factor of Fuer Elise is the maddening repetition of the E - D# tones of the opening: How many times do they repeat? Why? And why are they specifically articulated in rising pairs on page 2 while they had clearly begun on the upper note at the beginning?
To begin with a spelling lesson: In German "Th" functions as a single letter, and E-flat is called "Ess."
So we get, on page 2, (Th) E / (r) E / Ess (enharmonically D#) / E, her name. Would she have noticed?
Have you noticed?
Doesn't it explain why so much of the piece has to be played softly?
At any rate, the young man was right: the name is not at the beginning.