There have been proposals to generalize the notion of "measure zero sets" to "higher reals", that is, to subsets of ${}^\kappa 2$ for uncountable cardinals $\kappa$, typicallyoften inaccessible or weakly compact. (EDIT: sometimes also singular strong limits)
Shelah: A parallel to the null ideal for inaccessible $\lambda$, also here
Friedman-Laguzzi: A null ideal for inaccessibles.
Once you have the notion of a "higher null set", you can define "higher random" to mean "not an element of this or that null set" (for example: definable null set, etc)
(I prefer the adjective "higher" over "generalized", as it is more specific.)