It was Turing's paper, in which, among many other things, he showed that Turing machines were equivalent to lambda calculus, that convinced Godel that both Turing machines, and hence lambda calculus, were the right models of mechanical computation.
This was circa 1936, before Turing came to Princeton. So he was hardly unknown to mathematicians even then.
The Turing test and his codebreaking work are of course, well known. Of course, there are other achievements of Turing including a powerful version of the central limit theorem [2] , and "the chemical basis of morphogenesis" [3], to show that he was hardly incapable or obscure. Turing was an original genius, with a wide variety of original views that were later found to be far-reaching.
[1] https://www.jstor.org/stable/2371045?origin=crossref&seq=1#m...
[2] https://www.jstor.org/stable/2974762?seq=1#metadata_info_tab...
[3] https://en.wikipedia.org/wiki/The_Chemical_Basis_of_Morphoge...