In the Lagrangian formulation of Classical Mechanics, we have the freedom to add a total time derivative of an arbitrary function $\Lambda$ to the Lagrangian:
$$ L \to L + \frac{d \Lambda}{dt} . $$
Does this symmetry of the Lagrangian have any particular name?
Some authors call that a gauge transformation of the Lagrangian function. Others, don't give any specific name and may object to the previous one.
For a reference for the former denomination, see for instance F. Scheck, Mechanics, Springer, 2010.
I would simply call the operation$^1$ $$L(q,\dot{q},\ldots, q^{(N)},t)\quad \longrightarrow \quad L(q,\dot{q},\ldots, q^{(N)},t) \quad+\quad \frac{dF(q,\dot{q},\ldots, q^{(N-1)},t)}{dt}\tag{1}$$ for "adding a total derivative term to the Lagrangian", nothing else.
A quasi-symmetry transformation or a gauge transformation are by definition specified at the level of the fundamental variables of the theory (in this case, the $q$s and $t$). The operation (1) doesn't in general fulfill this.
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$^1$ The operation (1) should be amended with a prescription for the possible new boundary conditions (BCs). Note that the operation (1) [with the new BCs] may render the functional/variational derivative of the action $S$ ill-defined. However, if the functional derivative $\frac{\delta S}{\delta q}$ exists both before and after the operation (1), it is unchanged, cf. e.g. this Phys.SE post.
