I've been able to come up with a number of categories. Here are three of them as a partial answer (ASCII art because I'm having trouble figuring out the current state of LaTeX tables on SE):
$ \begingroup \small \newcommand \row[8] {#1&\sf #2&\sf #3&\sf #4&\sf #5&\sf #6&\sf #7&\sf #8} \begin{array}{l|ccccccc} \row{ }{Cu}{Ga}{Ru}{Lu}{Tl}{Po}{Pa}\\ \hline \row{\textsf{named after a country}^*}{ }{ X}{ X}{ }{ }{ X}{ }\\ \row{\textsf{prime atomic number} }{ X}{ X}{ }{ X}{ }{ }{ }\\ \row{\textsf{post-transition metal} }{ }{ X}{ }{ }{ X}{ X}{ }\\ \end{array} \endgroup $
* a recognized country at the time of the element's naming
