Understanding motivation/background for the name "infinitesimal generator" of a continuous semigroup This is soft question in the sense that I am looking to understand/motivate the naming behind the "infinitesimal generator" of a continuous semigroup. Namely, if we have a family of operators $\{U_t\}_{t \geq 0}$ s.t. $U_0 = I, U_{t+s} = U_t\circ U_s$ and the family is at least strongly continuous over a Hilbert/Banach space, why is the mapping $H\psi \equiv \lim_{t\to 0}\frac{i}{t}(U_t\psi - \psi)$ (in the quantum setting) called an infinitesimal generator? What, if anything, does it "generate"?
 A: The terminology comes from the branch of mathematics concerning Lie Groups and Lie Algebras.
In particular, we are searching for a matrix $X$ such that $\exp\left(tX\right) = U_t$. Here $\exp$ is the exponential map of Lie theory.
Now $X$ is an infinestimal generator of the family $U_t$ in the sense that $U_t$ are the operators traced out by the path that $X$ is pointing to.
In this case, the $X$ would be found by differentiating the path $t\mapsto U_t$ at $t=0$.
However, in your case there is a factor $i$ in the definition (I don't know why tbh, I'm not an expert) so I think you would rather have something like $U_t\psi=\exp\left(-itH\right)\psi$.

What, if anything, does it "generate"?

So to summarize, the (constant, i.e. not dependant on $t$) operator $H$ generates the (non-constant) operators $U_t$ via $U_t=\exp(-itH)$.
A: My answer applies to this quantum mechanical setting as well as to basically all linear (autonomous) differential equations (so the $-i$ in front of $H$ does not really make a major difference from the heuristical point of view). Much of what I am writing is formal, and can be made rigorous depending on the context, but it is really not essential to focus on this to begin answering your question.

As from another answer, $H$ is the operator such that $U_t=e^{-itH}$. In other words, if you take $\psi(t)=U_t\psi_0$, that is supposed to be the evolution of the state in your physical system, then $\psi(t)$ is a solution to the differential equation
$$ \left\{\begin{aligned} & \frac{d}{dt}\psi(t)=-iH\psi(t), \\ &\psi(0)=\psi_0 \end{aligned}\right. $$
In this setting, $H$ is the ‘vector field' that is telling you how $\psi$ is evolving in time. It tells you the law that governs the evolution of the state $\psi$ over time, and in principle uniquely determines all the future states $\psi(t)$, $t>0$ given the initial datum $\psi_0$. In this sense (to answer your question) it generates the flow $U_t$.
It is called "infinitesimal" because it only tells you what is the time derivative of your solution, and to recover the actual flow $U_t$ even for short times, you need to solve a differential equation. In the same way, for instance, the gravitational force between planets is the only thing that governs how planets move over time, but it is a long way from there to showing that planets' orbits are ellipses (to be fair, this specific example is actually a not very long exercise; but for a generic differential equation it is not possible in general to derive an explicit, exact formula for the solutions, and one simply proves that a solution exists and tries to study it qualitatively rather than quantitavely).
Heuristically, if you choose a very small time $h>0$, writing the first order Taylor expansion of your group, that is,
$$ e^{-ihH}\approx \mathbb 1 -ihH, $$
you see that if your state at time $t=0$ is $\psi_0$, then $\psi(t)$ at time $t=h$ is approximately
$$ \psi(h)\approx \psi_0-ihH\psi_0. $$
So that, if you approximate the evolution of $\psi$ with discrete time steps, the difference between one step and the next one is
$$ \psi(t+h)-\psi(t)\approx-ihH\psi(t), $$
Which means that infinitesimally (that is, at leading order), the operator $-iH$ is what tells you the difference between the state in the near future and the present state. It tells you how the state changes from the present moment to the 'next' time instant... Except there is no 'next' instant, as $t$ is a continuous variable (unlike other examples of deterministic dynamical systems which evolve in discrete time steps; think for instance about how a computer works, with an internal clock that makes the state of the internal memory change over time with a very high, but still finite frequency). So, $-iH$ is nothing but the "velocity" operator, that tells you how fast and in which direction your state is moving depending on the current position (note that "velocity" and "direction" have an abstract sense, and are meant to refer to the vector space in which your state $\psi$ lies, not e.g. to the physical 3D space or the like).
A: The notion of infinitesimal generator is used in a lot of domains (Probabilities, Partial differential equations, Lie Groups/Algebra, ...).
I would say to better understand the word "generator" in the expression "infinitesimal generator", one should first look at the non-infinitesimal case. Perhaps a concrete case is the case of a generator of random numbers. It gives, one by one, a list of random numbers. More generally, in probabilities, one sometimes works with Markov chains. On a finite set $J$, this correspond to just have a matrix $A= (A_{ij})_{(i,j)\in J^2}$ indicating the probability that if I am in the state $i$ I will then jump on the state $j$. Hence, starting from an initial state $v_0\in J$, the matrix "generates" a list of values $(v_k)_{k\in\mathbb N}$ such that $v_{k+1} = Av_k$.
The infinitesimal version is similar: $\partial_t u = A u$. The matrix or operator $A$ will give the direction of the variation of $u$ and so create (generate) the whole path. The information to create the path is in some sense stored in $A$. The word infinitesimal is of course here because derivatives are limit of infinitesimally small jumps, so this problem is the limit of $u_{t+\epsilon}-u_t = \epsilon\,A u_t$.
If $A$ is time independent, then the dynamic can be written $u(t) = e^{tA}u(0)$ and we see that the semigroup property directly in the exponentials, as well as the fact that $A = (e^{tA})'(0)$. In general, one can only deduce that the solution will be on the form of some semigroup with generator $A$
