If you change the names of the data files, that should not affect any application code including stored procedures. (But someone who is too smart for him- or herself might have written some code that breaks. Very unlikely, but I will not guarantee.)
Changing the database name... It is almost the same, but if there is cross-database access, or some people have written queries with three-part notation with hard-coded database names, they will still refer to the original database. I would say that this is not best practice, but it is a small risk that is not entirely ignorable.
If the database name is distinctive enough you can run this query to rule out this risk:
SELECT object_name(object_id) FROM sys.sql_modules
WHERE definition LIKE '%dbname%'