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In a Dan Brown book, the author claims that thoughts can interact with matter, and the entire field of study associated with this is called "Noetic Sciences".

Is this actually true? What is the veracity of this claim? Is there actually any proof behind this?

Thoughts consist of nerve impulses, and they most certainly can and do interact with matter- at least in the sense that nerve impulses can cause muscles to contract in the human body. But the context in which sensational claims are made for this (telekinesis, remote viewing, etc.) is something very different.

Claims that the mind can directly move or influence matter have been made for centuries and have gone by different names, but given the sensitivity of modern measurement tools, if any such effect did exist we would have detected it by now.

The veracity of these claims has not been demonstrated, and no proof exists that the effect is real. Furthermore, no mechanism for it has been furnished which does not in some way rely on "magic" i.e., something outside the realm of known physics.

There are a number of games available at this point which use a consumer-grade EEG machine to read brainwaves and control the action.

In order for you to play these games, your thoughts must either control or be your brainwaves and must affect some matter external to the body -- specifically the EEG sensors.

Strictly speaking there's no reason that specific circumstances couldn't result in a natural EEG sensor, either object(s) or other brains, but the effects are going to be subtle and highly dependent on the configuration and if such things were common in the general population we'd have more than folktales and anecdotes about it.

As far as practical use goes: further development of brainwave-scanning control systems for machines is far more likely to bear fruit. The energy of the human nervous system is quite detectable, but I know of no recorded instances where individuals could put all that much power into it. Developing an amplifier tuned to brainwaves might also be interesting, but the amount of power required to get significant effects from solid matter would make it a bit on the silly side and anything you could do with it you could probably do much more efficiently with more standard devices.

The OP asks these questions regarding "noetic sciences":

Is this actually true? What is the veracity of this claim? Is there actually any proof behind this?

Dean Radin, from The Institute of Noetic Studies, provides a select bibliography citing "peer-reviewed journal articles about psi (psychic) phenomena". The topics include "Mind-Matter Interactions" that the OP is interested in.

As a partial answer for whether there exists "any proof behind this" Radin's list makes it clear that evidence does exists for the claims.

The issue of whether it is true or not should be handled carefully. As with any science the results represent defeasible reasoning which Wikipedia describes as "rationally compelling, though not deductively valid." Calling the results of any science "true" may be an inappropriate use of the word.

Furthermore, although the data is real, one could use this data to support many different metaphysical commitments. It could be used to support panpsychism, or pantheism and even traditional theism. It could support something entirely new. There are aspects of these which are not likely compatible with the others.

One thing that the data doesn't support, unless there is further modification, is the belief that our universe is a computer simulation. Alan Turing recognized this as early as 1950 when he responded to the ninth objection to the imitation game:

I assume that the reader is familiar with the idea of extrasensory perception, and the
meaning of the four items of it, viz., telepathy, clairvoyance, precognition and
psychokinesis. These disturbing phenomena seem to deny all our usual scientific ideas.
How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming. It is very difficult to rearrange one's ideas so as to fit these new facts in. Once one has accepted them it does not seem a very big step to believe in ghosts and bogies. The idea that our bodies move simply according to the known laws of physics, together with some others not yet discovered but somewhat similar, would be one of the first to go.

This argument is to my mind quite a strong one. One can say in reply that many scientific theories seem to remain workable in practice, in spite of clashing with ESP; that in fact one can get along very nicely if one forgets about it. This is rather cold comfort, and one fears that thinking is just the kind of phenomenon where ESP may be especially relevant.

If one does not like the evidence for mind-matter interactions, one should not try to suppress it but adjust one's theories to explain it. If falsifiability is an appropriate approach to science this is how science improves.

If one does like the evidence, one should not assume it justifies more than it does.

Radin, Dean. Selected Psi Research Publications. Retrieved on July 1, 2019 at http://deanradin.com/evidence/evidence.htm

Turing, Alan (October 1950), "Computing Machinery and Intelligence", Mind, LIX (236): 433–460, doi:10.1093/mind/LIX.236.433

Wikipedia contributors. (2019, January 28). Defeasible reasoning. In Wikipedia, The Free Encyclopedia. Retrieved 15:20, July 1, 2019, from https://en.wikipedia.org/w/index.php?title=Defeasible_reasoning&oldid=880695250