Alphabetic / Shortalphabetic
Section 5.2 of the amsrefs documentation gives a list of fields that you can put inside the bibliographic entry. Among them you find
label: When the alphabetic or shortalphabetic options are used, amsrefs
will usually try to generate the label on its own. If necessary, you can
override the automatically generated label by specifying a label field.
See e.g.
\documentclass{article}
\usepackage[alphabetic]{amsrefs}
\begin{document}
Alan Sokal~\cite{Sokal96} recommends Bourbaki's
text~\cite{Bourbaki70} for a gentle introduction to set theory.
\begin{bibdiv}
\begin{biblist}
\bib{Bourbaki70}{book}{
title={Th\'eorie des ensembles},
author={Bourbaki, Nicolas},
date={1970},
publisher={Hermann},
label={SET}
}
\bib{Sokal96}{article}{
title={Trangressing the boundaries},
subtitle={Toward a transformative hermeneutics of quantum gravity},
author={Sokal, Alan},
journal={Social Text},
volume={46/47},
date={1996},
pages={217--252}
}
\end{biblist}
\end{bibdiv}
\end{document}
Numeric
The above doesn't work with the numeric option, since AMS takes the position that numeric keys should be linearly ordered from 1 to N in the bibliography.
What you need to do then is to split your bibliography into two separate biblists, one for the numeric items, and the other for the ones you want to name. Your choice of ordering the two lists influences which list is shown first in the bibliography.
Below we use the * form of the biblist environment to pass additional keys modifying the list from the default behavior, (as discussed in Section 3 of the amsrefs documentation). The first biblist is then alphabetic and so obeys the label instructions, and the second list is numeric (the default, since it was not otherwise specified when loading the package).
\documentclass{article}
\usepackage{amsrefs}
\begin{document}
Alan Sokal~\cite{Sokal96} recommends Bourbaki's
text~\cite{Bourbaki70} for a gentle introduction to set theory.
\begin{bibdiv}
\begin{biblist}*{labels={alphabetic}}
\bib{Bourbaki70}{book}{
title={Th\'eorie des ensembles},
author={Bourbaki, Nicolas},
date={1970},
publisher={Hermann},
label={SET}
}
\end{biblist}
\begin{biblist}
\bib{Sokal96}{article}{
title={Trangressing the boundaries},
subtitle={Toward a transformative hermeneutics of quantum gravity},
author={Sokal, Alan},
journal={Social Text},
volume={46/47},
date={1996},
pages={217--252}
}
\end{biblist}
\end{bibdiv}
\end{document}

amsrdoc.pdf, a copy can be found on CTAN: https://ctan.org/pkg/amsrefs?lang=en – Willie Wong Jun 14 '21 at 12:00