Now let me make sure I have a clear understanding of what abiogenesis means. It seems to be a field of study rather than a theory.
It's both, in the same way that evolution or quantum mechanics is a theory and there are scientists who study it.
Abiogensis seems to be about how life on earth in general could have arisen from inanimate matter rather than about how modern humans could have originated from inanimate matter.
Yes, no one has ever claimed that modern humans arose from inanimate matter. Modern humans are descended from other living things. However, if you trace our lineage back far enough, you'll get to those initial bits of genetic code and protein that were replicating themselves at the beginning.
"Mud theory" is obviously not a very accurate label. Is there an actual term for "the theory that modern humans originated from inanimate matter without intelligent intervention"?
Simply abiogenesis. There's no different theory for humans than any other organism. If you go back far enough, all things are related. Though, this theory would not be described that you have done it. Abiogenesis describes how cellular life evolved from molecules replicating themselves on some sort of substrate (what it was exactly is still being worked out). The ToE then describes how those initial unicellular organisms diversified into the millions of forms of life we see today.
Is there any evidence that we did in fact originate in that way?
Well, if you mean, is there fossil evidence on earth of acellular (nonviral) RNA or DNA replicating itself in mud? No. But, scientists have demonstrated that RNA can catalyze it's own replication on a clay substrate. They have found organics molecules such as amino acids on asteroids. Unless we can invent time travel, it's most likely that we will never come to 100% certainty about how live initially arose. However, the more we learn about the history of life and the Earth, we can come up with the most plausible explanation, and that will have to do. But saying "god did it" because we can be 100% sure is ridiculous. In the same way, we cannot know anything from the past with absolute certainty. For example, assume we knew that Lewis & Clark went across the continent, but they hadn't taken such extensive notes on their journey. We could piece together their most likely route based on evidence left of their campsites and things left behind. We may not get it exactly right (maybe they went right around some falls instead of left) but we can come up with the most likely explanation. Would you think that just because we couldn't figure it out perfectly that God must have picked them up and moved them across the continent?
For nothing else to be even remotely likely, it would take a lot of evidence that the mud theory actually occurred.
No it wouldn't. The possibilities of the alternatives are in no way dependent on the possibility of abiogenesis. Even if "life from mud" was wrong, it doesn't make "life from god" more plausible. There are many more naturalistic hypotheses and theories that are way more plausible than "a being of infinite power that does and does not exist in the universe and is able to affect it without leaving any evidence of action created life"
We have many intuitions in our life and the point is that many of these intuitions are wrong. The question is, are we going to test those intuitions? -Dan Ariely