Name for rooms overlooking a castle's great hall

Solution 1:

A lot depends upon the construction type and specific of the castle, but where such a room or passageway occurred, the most appropriate terminology could be:

Screens Passage - a passageway which connects to (and sometimes wraps around) the Great Hall

Arcade - a passageway with broad openings separated by columns ( or arches cut into a wall) with a significant overlook

Gallery - a covered passageway open to one side, or narrow balcony or platform running the length of a wall

Note that in most castles there was no such easy overlooking of the Great Hall for simple defensive reasons - it would have allowed too easy a line of attack on the resident noble at a high angle of inclination - ideal for archery.

There were in later centuries wooden screens hiding the passages to kitchens and other support / utility spaces, and in some cases wooden ladders or step up to private rooms on upper floors - and later even than that there were specific private dining and meeting rooms called withdrawing rooms which the resident noble might use for greater privacy and ease of use....

Hope this helps.