I'm using an iPhone 5 with the most recent iOS 6.02 update. I am also experiencing the lag issues, especially with messages.
I have plenty of storage left (12/16 GB available), little apps and videos etc., so I figured it has nothing to do with storage.
The problem seems to originate from messages so I figured it must have something to do with it. What could possibly cause lag in the messages? Hmm...
Ok I reasoned it down to 2 possiblities:
One, it might be the telcos fault,
Two, it might be too much messages is jamming up something somewhere.
The former seems unlikely since we assume service standards should have been maintained, and troubles would surface on a large scale affecting users of other phone brands. But since the problem is localised to the phone, hence, I tried to troubleshoot the latter.
One possible non-destructive fix you may try is to clear previous conversations, especially if the message thread gets way too long; like when you have been messaging to the same person for months and months (like your spouse or partner) and the thread like archives on and on and you did not clear it.
If the earlier conversations are insignificant, clear them. I did this and viola! I experienced much less lag and the phone returns to the lag-free state when I first bought it (ok occasionally some lag but much better now).
Give it a try and see if it works for you.
If it does, then it boils down to some basic housekeeping habits to clear earlier conversations and not jam up the message inbox with long threads, especially if the message contents need not be archived for future reference.