There are other languages where let used as a keyword before declaring a variable, such as BASIC and LISP (or Scheme), and I suppose it was taken from there. This is not an abbreviation; this is the normal English word "let", used to introduce a command, as in "Let there be light"; in mathematics, it is customary to declare a symbol in the same way as in "Let x be the unknown number of years that we are trying to calculate."
To answer your question a little more fully, though: in my opinion, there is nothing in this word, which makes it especially suitable for constants. It seems they made an arbitrary choice. var makes sense for a "variable" that can change (get?), so now they just need a different word and they choose let . Personally, I think const would be better.