Others have said some already, but for me:
The UI has to be monastic. No clutter at all, including no stats or resists or other information I could get from going to inventory. The only exception is current/max hp, xp and alt xp. Stuff like window selector, menu and other seldom-used windows are always off for me until I need them. Until the spell sets, my spell gem bar was always off too since all casts are done by hotkey.
The player gauge must be in the group window. This is probably the most important change for a priest UI.
While the popular buffs-with-their-name buff boxes are nice, having bard songs blinking to me are a distraction. I have just the icons in the shortbuffbox, shrunk to 16x16 and close to the gauges so I'm not looking elsewhere on the screen when I notice stuff blinking in the corner of my eye.
Four hotkeyed inventory slots: primary, secondary, top two general inventory slots. For swapping epic/proc hammer in primary and gvd/shield in secondary.
The hp gauges need to be high contrast and easy to gauge rate of damage taken over time. Part of this for me is making the target and group gauges the same size. I don't have %s or any cues for low life. I tried them and they were distracting. More often than not, I'd toss a small heal onto someone at 96% life during a lull and it was wasteful. I can understand it being a benefit for some, likely most.
Numerical mana % along with the gauge of course.
For a raiding cleric, a 10 min (+foci/aa) timer for Divine Intervention that you can get rid of when it's not needed. I spent an afternoon heavily modifying Arista's timer mod so that it fit unobtrusively into my UI and was just as unobtrusive for the majority of the time when it was off.
All gauges need to be in the same corner of the UI. When things get hairy, everything not in or immediately around the group window gets ignored.
Easy to manipulate chat windows are also nice for a raiding cleric. So you can create a new chat window for a ch rotation and get rid of it when it's not needed.
The loot window should have all 31 slots available without scrolling.
This is the one I use. With it you can pluck your epic from your corpse instantly without scrolling through to find it.
High visibility is crucial to a cleric. Being able to see the whole fight and potential problems in the surrounding area. With the need for high contrast gauges, this usually means a viewport or opaque group window background. One thing that helps is to keep everything orderly and sized enough to fit along the screen bottom as much as possible--keeping to a monastic style--to give an unobstructed rectangular viewport.