You can use a combination of list methods.

//start context
list<string> lst= new list<string>{'aa','bb','cc','dd'};
string var = 'cc';

//find the index of var
integer index; 
for(integer i =0;i< lst.size();i++){
 if(var==lst[i]) index=i;
}
//may want to throw an exception if index is still null

list<string> tempLst = lst.clone();//backup your collection
lst.clear(); //empty your collection

lst.add(tempLst[index]); //retrieve your desired record and put it on index 0
tempLst.remove(index); //remove your record from the old collection
lst.addAll(tempLst); //add the other data to your list 

system.debug(lst); /// |USER_DEBUG|[22]|DEBUG|(cc, aa, bb, dd)

But, logic like this might imply you should consider exploring the apex map collection type and its methods. Know you can use the map.values() property to still reference it as a list, or can directly iterate over a map in Visualforce too.