Don't have time to look at your file right now, but from the description, it appears that you are not renaming the value list but changing one of the values in your value list.
If so, redefining the value list does not change the data currently stored in this field. If there is data in that field that no longer matches one of the values in the list, it is still in the field, but is hidden by the check box format. (this is also true for radio buttons.)
You will need to do a batch update of your records changing the old value, where selected, to the new value and then you update your value list definition (or the table of values if it's a "use values from field" value list), to match.
Since a field formatted with multiple check boxes stores each selected value separated by a return and in the order that the user clicked the boxes, this is not a simple operation to do.
a) make a back up copy of your file just in case you make a mistake doing this update
b) enter find mode and click the check box for the value you intend to change. Perform your find
c) Temporarily remove the check box format from this field by returning it to "edit box" format. resize the field if you need to so that you can see all selected values.
d) click into this field
e) select replace field contents. Select the calculation option and use this expression:
Substitute ( Table::checkboxField ; "old value" ; "new Value" )
Note that I am assuming that the "old value" is not a substring of any other value that you might have in this field.
If the old value was "Apple" and another check box value was "Applesauce" for example, the above substitute would introduce errors into any record where Applesauce is selected. If that should be the case, don't do the above and post back here as I can show a more sophisticated expression that will avoid that problem.
Its a lot to take in (I am fairly new to Filemaker) there but I will look at this as soon as I can.