I asked Google Translate to do "There is always a choice" into Latin and it gave me "Est semper elegit" — is this correct? Is there a better way to put the same idea?
Google translate is almost always inaccurate when it comes to Latin (and, in my experience, most languages). Maybe as a per word dictionary, it is OK, but it cannot translate Latin for the life of it.
For example, in the sentence above, elegit is a verb. In English that would sound a bit like, "There is always chooses." Not exactly what you were going for.
The normal word for "choice" in Latin is optio (which is actually from where we get the word option).
Latin word order in cases like these is very loose, and ultimately it will be up to how it sounds. As a straightforward translation, you could have something like optio semper est or semper est optio or even est semper optio (of these three I prefer the first).
With a little more context concerning your quote, other options (pun not intended) become available. For example, most Latin proverbs of these type just leave off est entirely, e.g. in vino veritas ("in wine there is truth"). No verb needed. But it's difficult to give an optimal translation without seeing a bit how it will be used.
Hope this helps.