The earliest example that comes to mind is the shortening of the name Ted Nugent as The Nuge. To be clear, I'm less interested in the dropping of syllables at the end of his name, and more interested in the replacing of an initial word with "the".
A more recent example is Mountain Dew's slogan, "Do the Dew" (their re-branding as Mtn Dew is also interesting, but not what I'm getting at).
I've heard this formula applied colloquially, such as referring to Taco Bell as The Bell.
Would this be considered elision? If not, is there a name for this?