Check out the Replace Field Contents.
One of its options is to replace the contents of a specified field with a serial number. Simply pull up the set of record you need to update, sort them and then use this option (it's in the records menu) to assign your serial numbers.
Since this can change the values for a lot of records all in one step, you might want to save a back up copy before you try this one out just in case you get unexpected results.
Note that if this number is used as a key to link to related records, this tool will NOT update the related records. If this is the case, you would need to create and run a script to update it.