if the refresh of the system is used, the data state will not be saved. By refresh, you mean routing jump?

this is simple, just use localStorage directly, which is often used in my projects.

getInitialState: function() {
    var selectedOption = localStorage.getItem( 'SelectedOption' ) || 1;
    return {
        selectedOption: selectedOption
    };
},

setSelectedOption: function( option ) {
    localStorage.setItem( 'SelectedOption', option );
    this.setState( { selectedOption: option } );
}