On the Stanford Encyclopedia of Philosophy's article on Platonism in Metaphysics, the author writes that "Gödel's version of this view — and he seems to be alone in this — involves the idea that the mind is non-physical in some sense". In the footnotes to this article the author acknowledges that this interpretation of Gödel is controversial, but evidence can be found in his works published in 1951 and 1964. I think the works are "What is Cantor's Continuum Problem?" and "Some Basic Theorems on the Foundations of Mathematics and Their Implications". I read these but was unable to see any suggestion that the mind is non-physical. However, I may have accidentally missed evidence for this interpretation of Gödel. I would be grateful for any information or quotes that can explain where this interpreation comes from.
SEP itself refers to Platonism and Mathematical Intuition in Kurt Gödel's Thought by Parsons and On the Philosophical Development of Kurt Gödel by van Atten and Kennedy as the sources for this interpretation. Further discussion can be found in van Atten's book Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer and Conversations with Gödel chapter in Rucker's Infinity and the Mind.
However, not only is it controversial as an interpretation of Gödel's view, but even interpreting Parsons and others as presenting it would be controversial. Passages in 1951 and 1964 papers involve criticism of what is now called the computational theory of mind. More specifically, of its mechanistic version that Gödel attributed to Turing. In the 1963 Life Science publication of Mathematics Gödel's "disjunctive conclusion" was reported as follows:
"'Either mathematics is too big for the human mind,' he says, 'or the human mind is more than a machine.' He hopes to prove the latter".
There is, however, a leap in identifying non-computational (in the Church-Turing sense) with non-physical. Although it is true that physicalists typically adopt the computational theory of mind there is no reason why the physical can not go beyond the capabilities of Turing machines. Searle, for instance, famously opposed analogizing mind to computers and developed the Chinese Room thought experiment against the strong AI thesis. All while maintaining a form of materialism, see biological naturalism. Gödel's remarks were later developed into the Penrose-Lucas argument for uncomputable minds overcoming incompleteness, but in Penrose's hands, at least, it served as basis for conjecturing physical mechanisms of uncomputable behavior (objective collapse based on a speculative quantum gravity proposal).
It is possible, of course, that to Gödel non-computational mind meant non-physical mind, but neither those papers nor Parsons say anything about that. An indirect evidence for that would be Gödel's adoption of Husserl's phenomenology as the basis for his theory of intuition (and by extension, of mind's operation). Husserl himself officially suspended judgment on the question of realism (hence physicalism in particular), but it is clear that Gödel interprets phenomenology in the realist vein, and Husserl did adopt the title of transcendental idealism (albeit he then disavowed it late in life). Gödel also assimilates Husserl's eidoses to Plato's much more closely than Husserl would have liked. Gödel's reception of Husserl's and classical German idealisms are discussed in detail by van Atten and Kennedy, but they are more focused on objectivity of what is presented to the mind than on the nature of mind itself and "leave a discussion of Gödel's efforts on the question of minds and machines for another time".
The first relevant passage from Parsons is in footnote 16 on p. 52:
"However, the discussion in 1951 makes clear that Gödel regards the existence of recognition-transcendent truth as meaningful, since if the mathematical truths that the human mind can know can be generated by a Turing machine, then the proposition that this set is consistent would be a mathematical truth that we could not know."
The second is on p. 64:
"Thus in 1964, he emphasizes the fact that intuition gives rise to an "open series of extensions" of the axioms (p. 272), and of course the incompleteness theorem implies that any such series generated by a recursive rule would be incomplete and would, indeed, suggest a further reflection that would lead to a still stronger extension. Gödel interpreted these considerations by saying that the "mind, in its use, is not static, but constantly developing" (1972a, p. 306). This remark is directed against a mechanist view of mind such as Godel attributed to Turing. He explicitly offers the generation of new axioms of infinity in set theory as an example. It is interesting that the inexhaustibility of mathematics is used by Gödel both in drawing his analogy between perception and insight into mathematical axioms and in his critical discussion of mechanism".
"On this subject see 1951 (and Boolos's introductory note in CW III), Wang [24, pp. 324-326], and Wang . It would be beyond the scope of this paper to pursue this subject further. But it should be pointed out that what is needed for Gödel's case against mechanism is the inexhaustibility of our potential for acquiring mathematical knowledge. He himself makes clear in 1951 that that does not follow simply from the mathematical considerations such as the incompleteness."