Let me start by answering "Why do we look for a Theory of everything". The answer will partially answer the "need" question.
For each of us, from the time we open our eyes and maybe even before birth a succession of TOE s is vital for consciousness to connect with environment. We form consecutive maps of our observations and use them for predicting the next steps in our living experience. Like a developing numerical solution. Then we discover analogue methods which allow us not only to predict but also to control our environment.
So a TOE search is built in our cognition functions.
Then, as a human race we discovered mathematics that could map the world we observe simply and efficiently. This also gradually enlarged our view of the world, and at each level there were scientist proposing TOEs : from earth fire water air, to phlogiston and ether , geocentrism to heliocentrism, progress was slow because the mathematics was primitive.
With Newton and Maxwell mathematics was advanced by leagues and the effort for a mathematical TOE took off. Then came thermodynamics.
It took centuries for the application of these elegant proposals to appear useful for the man on the street, though at the time scientists thought they had the TOE.
Then came the expansion of our world view with the quantum mechanics revolution in the beginning of the 20th century. The man on the street is reaping the benefits of this. It took half a century for transistors to appear . In parallel special and general relativity modified kinematics and gravity.
The mathematical tools that developed in parallel were so powerful that for the first time, I think it was Kaluza-Klein, a unification of gravity and electromagnetism showed that the TOE might be expressed as one unified mathematical form, instead of a collection of axiomatic descriptions of disparate physical systems. And this is the road followed since then.
By the end of the 20th century most of the data that the standard model describes elegantly by unifying strong weak and electromagnetic forces in one mathematical format, had been gathered. Since then the goal for most theorists is to unify gravity in a TOE.
I want to stress the huge economic benefits of particle research to technology. The glaring example being this very webpage by which we are communicating with each other.Nevertheless nobody could have foreseen it. Most of the cost in the search for TOE is in the enormous, in size and number of people, experiments necessary to validate a TOE. It is very probable that these benefits will continue as long as there are physicists who will pursue the TOE. The economics will most probably make sense.
Will having a mathematical TOE predict unexpected effects that can be utilized as quantum mechanics generated the computer etc age? It is a gamble, probably yes, going by historical precedent.
The need is the inherent need of the human race to map predict and hope to control its environment.
Best Answer
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.