Atheists, what are the most convincing theist arguments you heard of?

[Brian37]

That just begs the question: how do you know whether the absolute is the ordinarily transparent apparatus of consciousness or some secret aspect of the world?

That actually sounds about right. Primal knowledge of one's self as personal existent (the apparatus of consciousness) and seeing the "world" something that actively participates with you in that experience (a secret aspect). This is pretty much what you will find in the historical literature of the faith traditions, as you can read below in a clipped quote of mine from another thread:

...I do not often refer to claims from special revelation but I think they have relevance....

“For now we see in a mirror, darkly…” – 1 Cor 13:12
"I am who I am." - Exodus 3:14
"For in Him we live and move and have our being..." - Acts 17:28
“You cannot see my face, for no one may see me and live.” – Ex 33:2
“And he was transfigured before them and His face shown like the sun and His clothes became white as light.” – Matt 17:2
“…Jesus Himself came up and walked with them, but they were kept from recognizing Him…then their eyes were opened and they recognized Him….” Luke 24:15,16

And for what it is worth:

“Angels have no notion or idea of time and space…” – Emanuel Swedenborg, HH 162
“If the doors of perception were cleansed everything would appear to Man as it is, Infinite.” – William Blake.
“The name that can be named is not the enduring and unchanging name.” – Lao-tzu

I mention these references both Scriptural and from mystical writings because all of them acknowledge a clear distinction between the nomenal and phenomenal. The categories of time and space and physicality are other than what we suppose. At the same time, the point of these sources  is that dispelling these illusions of fleshly incarnation reveals a divine order that is more fundamental (or spiritual) reality. They encourage people to look beyond the veil of illusions to see reality “as it really is.”

Yes, I have had an inkling of these experiences and that is mine alone. I did not mention it with the intent to convince you that my interpretation is correct. However, these types of experiences have been reported for millennia and systematically and successfully cultivated by practitioners from various times and places across cultures. I think that is something to which skeptics should give serious attention, not as proof, but as an pointer towards something available to them when and if they find themselves dissatisfied with where skepticism has ultimately led them.

"Personal experience" is not evidence. I used to see real shadows as a kid in my night time room that because of my ignorance and fear and age, my FALSE PERCEPTION lead me to think those shadows were bad ghosts or evil spirits. We do not doubt you had an "experience". We do doubt your perception of what you think you saw. 

People worldwide of all religions make all sorts of fantastic claims about spirits and ghosts and gods, and? So? Funny how when you listen to enough diverse stories the details of those similar "experiences" always match the social norms of the societies the claimer is born into.  A Muslim will chalk up their "experience" to Allah. A Buddhist will chalk their "experience" up to their superstitions It is a myth that Buddhists don't have their own versions of spirits or demons. Hindus will make similar claims and chalk them up to their versions of spirits or deities. 

If you really want to believe that Santa, and not your parents, put those presents under your tree, you will. There are no such things as ghosts or spirits or gods. People's flawed perceptions lead them to gap fill and something that may seem scary they end up believing those things out of ignorance. No not calling them stupid, but simply saying that most humans are ignorant of human psychology and how easily our own brains can fool us.

A majority of our species in the past had the false perception that the world was flat. too.