Several explanations have been proposed for the origin of the Copage/Coppage/Cuppage name:
1 Germanic Origins
The publication Burke's Irish Family Records has a listing for the Cuppage surname and traces the Irish (and later Canadian) family line to a Faustin Cuppaidge who moved from England to Ireland in 1604. It goes on to suggest that the name had Germanic roots although it is not clear on what evidence this is based. “The Coppage-Coppedge Family 1542-1955”, published in the US in 1955, mentions two brothers with the name "Coppetsch" who in the mid-seventeenth century studied at the University of Koenigsburg (now in Germany). However it goes on to point out that there is no evidence of any migration by Coppetsches to England and in any case there are instances of the surname in the UK well before this period.
2 Cup-page
Another suggestion is that the Cuppage surname comes from the term "Cup-Page", the official cupbearer to the king. However there is no evidence for this and while the appeal of such an explanation is obvious it would appear to be based on no more than wishful thinking.
3 Corbridge
The village of Corbridge in Northumberland is built on the site of the ancient Roman fort of Coriosopitum, part of the defences along Hadrians' Wall. However it is highly unlikely that the name "Corbridge" would transmute to "Coppage" as this would require both "R"s to be dropped. Furthermore, words ending with the common Saxon "bridge" tend to remain unchanged.
4) Coppedhegge Forest
In the late thirteenth century there was a small forest in Buckinghamshire called "Coppedhegge" near the border with Oxfordshire and now part of Bernwood Forest. Coppedhegge comes from the Anglo-Saxon "Copedehecg" and given that the Saxon terms "cop" and "Hecg" mean "top" and "forest" this suggests a "topped forest" or "forest on a rise". Only three inhabitants were recorded in the forest, possibly woodcutters, and it is possible that they bore or were given this surname. What lends credence to this last theory is the striking proximity of the forest to known clusters of bearers of the surname in the sixteenth and seventeenth centuries. The forest is very close to Aston Rowant in Oxfordshire (5 miles), Chaddleworth in Berkshire (15 miles), Leighton in Buckinghamshire (21 miles) and Cricklade in Wiltshire (25 miles). There are numerous records of Copages in all these four settlements.
So while we may never know for certain the origin of the Copage/Coppage/Coppedge/Cuppage surname the last of these four theories would seem to be by far the most likely.

The area surrounding Coppedhegge forest showing nearby clusters of Copages