Personally, I naively took it to mean a complete description of the physical universe, which does remain very vague, and may not be meaningful.
I can describe for you the mathematical form of a theory of everything:
Where # is a symbol which includes convoluted differential forms. This is the form all partial mathematical models of electromagnetism, mechanics, thermodynamics ... every type of mathematical model for physical systems has taken in the past and will appear in the future. They can be reduced to differential forms on the left and a zero on the right.
It will have its mathematical axioms that will make the solutions rigorous but it will have the physics postulates that have to be satisfied as the input in choosing a particular TOE from multitudes of similar ones.
The word "theory" in physics does not simply have the burden of mathematical rigorous existence. It could be very rigorous mathematically and irrelevant for the physics to be modeled. Mathematical theories become models for physical states.
The fact that it will be mathematically rigorous means that the solutions will describe correctly all known physical data and predict, given the boundary conditions, any new ones we could think about, in precise numbers. There is nothing vague about it.
We expect that the Standard Model, which describes almost completely all known up to now particle data, will naturally nest in the model of the TOE. There is nothing vague about this either. We expect gravity to be modeled naturally within TOE.
At the moment it seems that string theories offer all these options, but as there are thousands of possibilities, the final model has not been found yet, not even the class of models within string theories, which can be candidate for the embeddings necessary of the SM to assure consistency with existing data. If/when decided upon the predictions of the model will be tested for consistency with new data.