Hamiltonian formalism
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Prior to the dawn of Quantum Mechanics, there was Classical Mechanics. This was a theory initiated by the Italian polymath Galileo Galilei with his studies of kinematics and inertial frames of reference and later formalized by Sir Isaac Newton with his three well-known Laws. This Force-oriented theory of movement soon proved to be as useful as it was complex, since it requires to solve a system of up to three second order ODEs which are probably coupled. The need for a more modern, elegant and general approach was evident, and it was Joseph-Louis Lagrange who brought it to life (with the noteworthy mention of D'Alembert), and with it the advent of Theoretical Mechanics. Despite this great achievement, it would be William Rowan Hamilton who would accomplish the feat of reducing Classical Mechanics to a single first principle ("the path followed by an elementary particle is such that the action is stationary"), and introduced the Hamiltonian formalism.
In Quantum Physics, one works with probabilities rather than with certainties. Therefore, the formalism must be different than the Classical one: Erwin Schrödinger introduced the concept of "wavefunction" one can assign to a particle, which encodes all of the available information about the particle (this is a postulate). For a wavefunction to be a candidate to describe a quantum system, it must fulfill two conditions: (1) to be a solution to Schrödinger's equation and (2) to be normalizable (that is to say, to be square-integrable). Schrödinger's work was based on the Classical Hamilton-Jacobi theory, so it is not surprising that the operator we find in his time-independent equation is called, precisely, Hamiltonian.
$H\Psi = E\Psi$
where $H$ is the Hamiltonian operator, $\Psi$ is the spatial wavefunction and $E$ is the energy of the particle in state $\Psi$. As one can see, this is an eigenvalue problem for some differential operator $H$, which is defined as:
$H \equiv \frac{p²}{2m}+V$
In total resemblance to the Hamiltonian used in Classical Mechanics, but with the essential difference that the so-called observables ($x$ for position and $p$ for momentum) are now operators acting on functions of a Hilbert space.