Hi Eddie,

The best/ easiest way to go about this is to create a second date or date/time field for the last modified date for that specific field. Then in a simple process builder check if that field value has changed and if so populate the timestamp. Then you could create a formula field to come up with the score by checking the timestamp vs the current time and determining if it is 1-3 months old, 4-6 months old, etc. 

There is a programatic solution to this where you could run a scheduled batch class (Would have to be this unless you wait until the object is updated to run through a trigger or a button is pressed to launch the code via a lightning component. But a scheduled batch would do it automatically) to query the object and it's related history something like

SELECT Name, customfield__c, (SELECT OldValue, NewValue, CreatedDate FROM Histories WHERE Field = 'customfield__c' ORDER BY CreatedDate DESC) 
FROM foo__c WHERE customfield__c != null

And then checking the first index of the related history objects createddate
 

for(foo__c record : scope){
 if(record.Histories.size() != 0){
  Integer monthsBetween = 
  Date.valueOf(record.Histories[0].CreatedDate).monthsBetween(System.Today());
  if(monthsBetween <= 3){
  record.Score__c = 5;
  }
  else if(monthsBetween >3 && monthsBetween <=6){
  record.Score__c = 4;
  }
  //etc.
 }
}
update scope;

Now the code above hasn't been tested so I can't guarantee functionality but that was just a rough draft of something I think could work. Again, the first solution is simpler and easier to maintain and since it is a formula field once the timestamp is updated it will automatically recalculate rather than a scheduled batch. 

If this answers your question please mark as best answer!
Regards,
Tyler