It seems that the usual way in modern mathematics is that one author is investigating a particular mathematical structure in a research paper, and then other authors reference this structure with the name of the original author attached. Usually, important theorems are named after people who published them (cf. also Stigler's Law). This also results in important mathematical objects from the theorem to get the same name.
David Hilbert developed a theory of quadratic forms of infinitely many unknowns in a series of notes in the years 1906 to 1910, essentially laying the foundation of selfadjoint operators in Hilbert spaces. But the methods involved do not make use of Hilbert spaces (the only one is $\ell^2$). It's told () that Hilbert asked the following, after a talk by Herrmann Weyl (translation by me):
Weyl, there's one thing you need to explain to me: What's a Hilbert's space? I did not understand this.
Stefan Banach contributed in an essential way to the examination of Banach spaces; his dissertation was printed in 1922. The term "Banach spaces" was introduced by Fréchet in 1928; in a monography from 1932, Banach himself uses the term "spaces of type (B)".
Similarly, Hirotugu Akaike introduced in his 1974 paper "A new look at the statistical model identification " an information criterion called "An Information Criterion" and abreviated it with AIC. Conveniently, this was backronymed to "Akaike's Information Criterion" rather quickly, e.g. in N. Sugiura's "Further analysis of the data by Akaike’s information criterion and the finite corrections" from 1978.
: Dirk Werner: Funktionalanalysis, Chapter V: Hilbert Spaces, Sec. 7: Remarks (German)