You can just push the value to a cookie using JavaScript or jQuery and then let PHP retrieve the value from that cookie like this:
JavaScript + PHP:
/* JavaScript */
const switch = document.querySelector(".avflipswitch")[0];
switch.addEventListener("change", function(){
let blankTar = "_blank";
let selfTar = "_self";
if(this.checked){
document.cookie = "target =" + blankTar;
window.location.reload();
}
else{
document.cookie = "target =" + selfTar;
window.location.reload();
}
}
/* PHP */
<gcse:searchresults-only linktarget="<?php echo $_COOKIE['target']; ?>"></gcse:searchresults-only>
jQuery + PHP:
/* jQuery */
$('.avflipswitch').on("change", function (){
let blankTar = "_blank";
let selfTar = "_self";
if(this.checked){
document.cookie = "target =" + blankTar;
window.location.reload();
}
else{
document.cookie = "target =" + selfTar;
window.location.reload();
}
});
/* PHP */
<gcse:searchresults-only linktarget="<?php echo $_COOKIE['target']; ?>"></gcse:searchresults-only>