Is it possible for a 16 year-old to choose whether she wants to live with her parents or not?

I can assume from your question that your girlfriend's father's ex-wife is not her Mother but someone she obviously cares for and trusts.  Who "sent" her to live with her?  The Court? 

Generally, if during a determination by the court of who should have phyical custody of a child of the age of 16 as between parents the Court will listen to the 16 year old and weigh their opinion.  But this is not the case here.  The ex-wife is not her Mother.  Although there may be an emotional attachment there is not a legal attachment.

If the ex-wife is willing to have your girlfriend live with her she would need to petition the court for physical custody.  Custody determinations are made in the best interests of the child.  The court will probably appoint an attorney for your girlfriend and your girl friend will be heard.  Your girlfriend may have other options being 16 such as emancipation but that is a huge step and not one that should be entered in to lightly.  

An attorney in your area who can look at all the facts in this situation should be consulted.   Check with your state social services department about free/low cost legal services that may be available to minors. Good luck.