Use a SP to Archive/Revision related records in several related tables? I have a database for an application that has a complicated relational table structure. Is there a good way to create revisions of the data, or archive the data before a user makes changes? 

Ideally I would want a stored procedure that would save the state of all the related data when ran, insomuch that when a field is added to any of the related tables, the stored procedure would not miss it.
Creating such a stored procedure is probably going to be complex and error prone. That's why there are temporal databases or temporal features in a database. If you really wanted to do this with a stored procedure, you can have a duplicate schema with a version number column added to all tables. 

That way, on each write operation, you can first call the stored procedure which does an INSERT INTO.... SELECT using the same WHERE clause as your write operation. The stored procedure is best initiated by the app which runs the stored procedure before the actual write (UPDATE/DELETE) statement. 

If you use triggers, it can be very tricky just figuring out which table(s) to host the trigger and making sure you don't get a trigger storm.

Far easier path is to implement something like Temporal tables in SQL Server 2016: https://msdn.microsoft.com/en-us/library/dn935015.aspx. Takes a bit of work to setup but it's only somewhat tedious work, not very complex work. If you want to keep this history around for a long time, you can even stretch the temporal history table so you don't have to deal with administration and storage challenges with a massive history table.