Other answers are most welcome, but here I will outline the process through which I went to find a reference that you can cite for a hydrogen pseudopotential. Hopefully this will help you find references for other other elements, and to dig deeper on your own in cases where references for these types of things seem harder to find.
I followed the URL in your question, and as you said, it lead to a link to the ABINIT database. From there I clicked the link that said "Atomic data / pseudopotentials", then the link that said "Recommended norm-conserving pseudo potentials" which got me to this page. By clicking the H in the first periodic table shown, I obtained a file called
H.psp8 which says in a comment:
#While it is not required under the terms of the GNU GPL, it is
#suggested that you cite D. R. Hamann, Phys. Rev. B 88, 085117 (2013)
#in any publication utilizing these pseudopotentials.
Therefore, for that specific norm-conserving pseudo-potential, you can cite D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
The numbers in there, such as
-5.26468 don't show up in, for example, the PSF file that we get by clicking the
H button directly on the site whose link you provided.
So if you want, you may have to apply similar techniques of digging into things in order to find a reference for the data in that PSF file. Also note though, that one of the ABINIT pages that I previously mentioned but for which I didn't yet provide a link (here it is) says:
"To use ABINIT in specific conditions it is recommended to produce
your own atomic data sets.
This is the case, for example, if one wants to make simulations in
specific thermodynamic conditions, or use specific
exchange-correlation functionals, etc...
- Norm-conserving pseudo potential generators
- PAW atomic data file generators
If you follow that advice, you can generate your own data, and cite the program (from any one of the links in the quote directly above). Citing these programs seems a lot more straightforward than finding the citations for data given in databases that don't make it clear where they got the data (though generating one's own data is not the most convenient way to get a pseudopotential). While looking through all this, I've found some pesudopotential files in which there's no citation provided and not even anything close to a clue: this, along with the
comment above by Hamann that says "While it is not required under the terms of the GNU GPL, it is suggested that you cite [...]" suggests that the authors of these pseudopotentials are not so fussed about getting citations for use of their pseudopotentials, and the lack of easily-accessible citation information given by SIESTA and ABINIT suggests that the "common practice" of citing this type of pseudopotential data is quite different from the common practice for citing basis sets, as you can see MOLPRO's equivalent online database for basis sets gives references, for example the cc-pVDZ basis set for hydrogen is here along with the literature citation:
"H : T.H. Dunning, Jr. J. Chem. Phys. 90, 1007 (1989)."
Note though, that even when using basis sets, a lot of people don't give citations for the original papers from which the basis set came, such as the Dunning paper mentioned directly above, but this is usually because they don't know any better. You are doing the right thing by trying to find citations for the pseudopotentials you're using.