How do I store an array in localStorage?

localStorage only supports strings. Use JSON.stringify() and JSON.parse().

var names = [];
names[0] = prompt("New member name?");
localStorage.setItem("names", JSON.stringify(names));

//...
var storedNames = JSON.parse(localStorage.getItem("names"));

You can also use direct access to set/get item:

localstorage.names = JSON.stringify(names);
var storedNames = JSON.parse(localStorage.names);