Hamilton's equation reads $$ \frac{d}{dt} F = \{ F,H\} \, .$$ In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time evolution of $F$. In other words $H$ generates temportal shifts $t \to t +dt$.

The function $F$ over phase space describes a conserved quantitiy if $$ \frac{d}{dt} F = \{ F,H\} =0 \, .$$ Nother's theorem now exploits that the Poisson bracket is antisymmetric $$ \{ A,B\} = - \{ B,A\} .$$ Therefore we can reverse the role of the two functions in the Poisson bracket above $$ \{ F,H\} =0 \quad \leftrightarrow \quad \{ H,F\} =0 \,. $$ In words, this second equation tells us that for any conserved quantity $F$, its action on the Hamiltonian $H$ is zero. In other words, $F$ generates as symmetry. This is exactly Noether's theorem.

But usually, we argue that only the Lagrangian has to be invariant. The Hamiltonian can change under symmetries like boosts which increase the potential energy. (While the Lagrangian is a scalar, the Hamiltonian is only one component of the energy-momentum vector and therefore, there is no reason why it should be invariant.)

So why exactly do we find in the Hamiltonian version of Noether's theorem that the Hamiltonian remains invariant under symmetry transformmations?