Mathematician's names in structures.

In some cases is totally whimsical. Funny quote for Rudin, Functional Analysis about Tychonoff's Theorem:

"A. Tychonoff proved this for Cartesian products of intervals (Math. Ann., vol. 1 02, pp. 544--561, 1930) and used it to construct what is now known as the Čech (or Stone-Čech) compactification of a completely regular space. E. Čech (Ann. Math., vol. 38, pp. 823-844, 1937 ; especially p. 830) proved the general case of the theorem and studied properties of the compactification. Thus it appears that Čech proved the Tychonoff theorem, whereas Tychonoff found the Čech compactification-a good illustration of the historical reliability of mathematical nomenclature."

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.

For instance:

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 ([1]) 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.

[1]: Dirk Werner: Funktionalanalysis, Chapter V: Hilbert Spaces, Sec. 7: Remarks (German)

I think no one ever decides to name something after himself. Often enough person A first does some pioneering work, then person B comes along and identifies the proper space/notion/object underlying the work of person A and expands further the theory, and proposes to name it after A.

Or A comes up with some really important notion, then everyone cites A's results/notion and by consensus it becomes A's space/graph/theorem or whatever.