Ask for a disc with firmware 12116A for the TX-SR876.
They should reply with a form to sign...
The 3010 is the descendant that occupies the same spot in the lineup, but it is difficult to make a hard comparison because the feature sets of even lower level models have exceeded much of what we are familiar with in our 876s.
The 1010 also maintains THX Ultra certification and has the same HQV Vida video processor as the 3010. Both the 3010 and 1010 allow their Sub HDMI out to be used as an HDMI Zone 2 output... something that has been on the wish list for many folks.
The 818 is also a very interesting model... having a somewhat less robust amp section (THX Select certified) and a different (but still considered premium) video processor, but with a significantly lower price than either the 3010 or 1010. The 818 only has seven on-board amp channels vs. nine for the other two... however, it includes Audyssey MultEQ XT32 just like its big brothers have. Many folks consider this a huge step up from the MultEQ XT processing in lesser/older models.
I would think that an 876 owner would be primarily looking at these three models and weighing their features/cost against their own needs/budget.