A space $X$ is discretely generated (DG) if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a discrete set $D \subset A$ such that $x \in \overline{D}$. 

This is a convergence type property (and in fact, Fréchet-Urysohn and even radial spaces are DG) but it's pretty weak: every scattered space is DG, every compact space of countable tightness is DG in ZFC and every monotonically normal space is DG (see <cite authors="Dow, A.; Tkachenko, M.G.; Tkachuk, V.V.; Wilson, R.G.">_Dow, A.; Tkachenko, M.G.; Tkachuk, V.V.; Wilson, R.G._, Topologies generated by discrete subspaces, Glas. Mat., III. Ser. 37, No.1, 187-210 (2002). [ZBL1009.54005](https://zbmath.org/?q=an:1009.54005).</cite>).

There are examples of DG spaces with a non-DG square (see  <cite authors="Murtinová, Eva">_Murtinová, Eva_, [**On products of discretely generated spaces**](http://dx.doi.org/10.1016/j.topol.2005.11.014), Topology Appl. 153, No. 18, 3402-3408 (2006). [ZBL1107.54006](https://zbmath.org/?q=an:1107.54006).</cite>), but in a few notable special case discrete generability is indeed preserved in box products:

>> THEOREM (Tkachuk, 2012): Every box product of monotonically normal spaces is discretely generated.

>>THEOREM (Barriga-Acosta and Hernández-Hernández, 2016): Every box product of regular first-countable spaces is discretely generated.