In the hierarchy of theories, first comes hamiltonian theory, from which one deduces kinetics theory, and at last thermodynamics and fluid theories. From a kinetics point of view, entropy and temperature are macroscopic parameters with no equivalent at the microscopic level. However, thermodynamical entropy and information entropy being isomorph, there has been a general trend to generalize thermodynamical properties to non Hamiltonian systems and use entropy as a fundamental property. Prigogine had tried at the end of its life to give entropy a fundamental status and to redefine hamiltonian mechanics, starting from entropy. Does anyone know if there has been any successful attempts in this direction ?
I'll expand my comment into a partial answer. GENERIC framework is a generalization of Hamiltonian mechanics to include irreversible processes. If Hamiltonian mechanics features only energy as a function that produces the dynamics of the system, GENERIC introduces entropy alongside.
If you look at the GENERIC, you'll see that without that entropy part it is just Hamiltonian equations.
As mechanics itself tells little about the form of Hamiltonian, so GENERIC doesn't tell much about the form of entropy, you should know it beforehand. Ultimately it is just a general form of equations governing a classical system, just like Hamiltonian equations govern classical reversible systems. Unfortunately energy and entropy are not the only properties required to describe the system, one has also to introduce an additional linear operator $M$, which embrace constitutive equations like for example Fourier's law.
Books, that might help. I'm not sure there is much about GENERIC written. Apart from the already mentioned
there are two books with a review chapter on GENERIC by almost the same authors:
These are really nice and present a review of various (quite a lot!) aspects and approaches to the subject of modern thermodynamics.
To sum up, GENERIC is exactly "a modification of the dynamical equations that would take into account the entropy".
Now to the part "rebuild mechanics from scratch with entropy at the microscopic level being taken into account". I don't know any, but I wasn't looking for. Non-Hamiltonian mechanics is actually used in practice in molecular dynamics. For example Nosé–Hoover thermostat efficiently modifies microscopic dynamics of molecules introducing some sort of friction (which can be positive or negative). But it couldn't be viewed as some fundamental theory, it is rather an engineering trick to make a small system behave as desired.