Tom,
Great topic. I am not sure I know either, and I agree that like anything else, there are rules - often broken - of thumb (ROBOTS) but not that hard and fast a rule.
The par 3's I consider great probably have a great setting. Since you can design the hole to be viewed in its entirety, there is no reason it can't be visually spectacular. Naturally spectacular like Terry Hatcher?) always better than artifically spectacular.....but artifical, with bunkers, etc. not bad either.
I have always believed 3 pars are less strategic, as there is no shot relationship, even though there is strategy in shot location, shot pattern, etc. As such, I generally feel they are good places to test specific skills or concept shots, like CBMac's Biaritz, Redan, Short, etc. To use another showbiz analogy, Curly in City Slickers says "Theres just one thing.....and you got to find it" That may be different for everyone, but it must be memorable for something specific - for both the visuals AND the type of shot.
In modern design, I think an ultra long par 3 can be great, if only to test a long iron shot. On two and three shot holes, the player can lay back to full wedge distance, so its also a chance to create the half wedge shot, so ultra short holes, like PB 7 are great too.
I agree generally that the margin for errors can be less than on similar greens for par 4 holes, given the gca control of the shot from the tee. However, I think there are too many "straightforward" par 3 holes that are accuarcy only tests.
I also agree that the great par 3 must play differently every day, rather than be automatic to be great, in most cases. This can be through varied tees to a smallish green or through a large green from limited tees, or a large green with large, varied tees!
Someone mentioned they might be considered as a set..... I suppose that four very good par 3's help make a great golf course, but certainly, a great par 3 could stand as part of an otherwise lackluster course, or for that matter, stand out on any course.
If I really come up with some good thoughts, I will post again!