The Staff of Herding's specific reagents are rare drops, specifically Act 2's Liquid Rainbow and Act 3's Gibbering Gemstone. My friends and I have already spent hours farming these to no avail. Is there anything I can do to influence these drops? Perhaps Magic Find or changing the difficulty? Is there any way to reliably predict whether the dungeon or chest/monster has spawned from other nearby features? (Is the Liquid Rainbow always in the front of the cave? Is there always only one Unique in Caverns 2?)
I tend to favor +% MF on my gear, and I don't believe that it has a large impact (if any) on the spawn rates of these items. They're not magical, and there are only one or two which are actually monster drops, so more MF% is unlikely to make any difference.
Part of the problem is just that you have to get "lucky" with the map creation, which means that you're going to be exiting and reloading the game frequently. Being on the right quest/checkpoint/waypoint/etc can make this somewhat faster.
For the most part, what I'd recommend is following the per-item guide I've laid out in this answer:
In each item's case, I've given the specific conditions by which you can tell whether or not the dungeon or item in question has spawned. I think for each of these I spent about 2-3 minutes determining failure, and tried probably less than 15-20 times to get them (with the exception of the Plans, which are relatively painful per-run but have fewer random elements involved).
One trick I have seen is that if you're farming these items along with friends, have each of you in your own instance of the game - if you manage to find the chest/lootable object/monster/etc, have your friends join your game. When you open the object, you'll each get a copy of the item. Since you're essentially rolling multiple random maps or what have you at once, your chances are increased significantly. Make sure you don't expose the loot prior to your friends joining, as they won't be able to see or interact with the object if it has already spawned.