I think it has more to do with whether or not the piece was commissioned or written in honor of someone. I'm thinking specifically of Beethoven's Symphony No. 3 in E-Flat Major, which originally bore the title "Buonaparte" after Napoleon. When Napoleon declared himself Emperor in 1804, Beethoven furiously renamed the work "Sinfonia Eroica," or "Heroic Symphony" in protest against the act.

To use the example you cited, "Für Elise" was written for one of a handful of possible women whom Beethoven is said to have admired, though no one is really certain who precisely it was dedicated to. It's technical name, however, is Bagatelle No. 25 in A Minor.

Sometimes, I think it just depend on whether or not the composer had a specific narrative in mind for the work, such as Chopin's "Raindrop Prelude," which allegedly came to him in a dream of the death of his lover's son.

Edited to fix a minor/major issue

Traditionally, pieces from the Classical music tradition were named after the genre of the piece - such as Concerto, Symphony, Suite, etc. - and then the number that the composer had written, the key of the piece, and then if they were published, they were assigned an opus number (or a Deustch, Kochel, etc. depending on the composer.) Differentiating the genres was important, as the difference between sonatas, symphonies, concertos, tone poems, suites, variations, etc. was vast, in formal structure, instrumentation, (generally) scope of the piece, etc. Musicians in the 18thC were often considered (and considered themselves) as tradesmen, not artists as we know it, which also contributed to the more workaday names given to much of the music written during that time.

Fur Elise, as /u/kittydentures pointed out, was actually called Bagatelle No. 25 in A minor. In this case, the form of the piece is a bagatelle (a type of dance), the 25th that Beethoven had written, and in the key of A minor. Often, nicknames associated with pieces were not even chosen by the composer - to stick with Beethoven, both The Tempest (Sonata Op. 31 No. 2 in D minor) and The Moonlight (Sonata Op. 27 No. 2 in C# minor) acquired those names independently of the composer.

Another reason for naming pieces this way, especially with the rise of 'hausmusik' (music for the home) in the 18th and especially the 19th centuries, was so that customers knew what they were buying. Franz Joseph Haydn's String Quartets were published as that (and not by the names they later acquired) so that when they were bought, people knew exactly what they would need to play it - in this case, two violins, a viola, and a violoncello.

This did eventually start to fall by the wayside, with the rise of programmatic music throughout the 19thC (such as Hector Berlioz's Harold in Italy, Franz Liszt's Years of Pilgrimage, or Richard Strauss' Also Sprach Zarathustra) but certainly never vanished.