Assuming that you are saving the value of the previously selected item, you can pre-select it this way
code
var lsvPrevSelectedItemVal = localStorage.getItem('custPrevSelValueFromDb');
// Remember to change the select component name
Apperyio("mobileselectmenu_20").val(lsvPrevSelectedItemVal).selectmenu('refresh');
/code
You can call this code after the REST Success call so that it executes it after the Select component has been populated with values. As an added measure you may want to do some error handling in case the previously selected item is no longer available / deleted etc