In the web world, the usual thing to do is to either save data in Cookies or Sessions. Cookies are stored in the user's machine until its expiration date is met, while Sessions usually have a shorter time span (depends usually on the server's session time to live).
One approach, if the user can leave and come back at a later point, (even after a restart), is to save it to a cookie. You can use setcookie in PHP for this, and serialize the data.
Example:
setcookie("UserFormData", serialize($_POST), time()+3600); /* expire in 1 hour */