DAOs still have shareholders - any AI they may have seems like it is limited to operational decision making. The DAO may own assets on behalf of its shareholders but the AI doesn't own anything.
Until AIs have recognized rights that include property rights, AIs cannot own anything.
Patents can be assigned to corporations, but corporations cannot be named inventors.
Do you mean AI find things? Current AIs are just a complex algorithm for searching or approximating stuff. You might argue that some algorithms "learn" but that is just an algorithm that is adjusting some weights based on some algorithm someone created. AI is a tool, tools don't invent things. Maybe future AI like the ones in SciFi could invent things but for now we don't know that is even possible.
"The laws of the United States hold that a legal entity (like a corporation or non-profit organization) shall be treated under the law as a person except when otherwise noted. This rule of construction is specified in 1 U.S.C. §1 (United States Code), which states:
In determining the meaning of any Act of Congress, unless the context indicates otherwise—
the words "person" and "whoever" include corporations, companies, associations, firms, partnerships, societies, and joint stock companies, as well as individuals;"
If a corporation can own patents and have legal personhood, then you might get to the point where an AI is the corporation, and thereby could own patents. I am thinking of the DAO (Distributed Autonomous Corporation). I am also thinking of the Cat series of novels by Joan D. Vinge, which contain an AI run corporation [3].
Another consideration is that we are seeing AIs invent things now in the fields of medicine, material science, music and perhaps others. If AIs own these inventions then whoever owns the AI owns the inventions.
Given the DAO, if an AI can own assets then I don't see any way an AI cannot own a patent.
Yet another angle though is liability. If an asset owned by the AI is found to be the cause of someone getting injured, then who is at fault? Who has to pay damages?
[1] https://en.wikipedia.org/wiki/Corporate_personhood#Case_law_...
[2] https://en.wikipedia.org/wiki/The_DAO_(organization)
[3] https://en.wikipedia.org/wiki/Joan_D._Vinge#Cat