@oristarA,
The name Elmo is not associated with the tree elm.
But, because there is a name that starts with elm, they just tagged the "o" on the end to give it a name.
Also, there is the Disney movie "Saving Nemo", and "Saving Elmo" is a play on that.
Not to mention the popular Sesame Street character Elmo.
So, in all, they were just being clever.