So this site is wordpress with bbpress for forums right? Do you roll your own database with that?
Intuition says its likely a database column that needs and index ... but dont know how this is hosted, maybe you buy a certain amount of bandwidth... but that does not align with the consistent latencies .. some times a day it should be faster than others... doesnt seem to be the case. So most probable is that database history has caught up with a lacking query / missing index
Is my bet.
(been at this since I was ~10)