There have been proposals to generated the notion of "measure zero sets" to "higher reals", that is, to subsets of ${}^\kappa 2$ for uncountable cardinals $\kappa$, typically inaccessible or weakly compact: 

- Shelah: [A parallel to the null ideal for inaccessible $\lambda$](https://shelah.logic.at/papers/1004/), also [here](https://arxiv.org/abs/1202.5799)

- Friedman-Laguzzi: [A null ideal for inaccessibles](https://arxiv.org/abs/2004.11975).

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.)