Who, besides Hamilton, was instrumental in discovering at least part of the so-called Hamilton principle?
$$\delta \int_{a}^{b}L(q,\dot q ,t )dt$$
where $ L=T-V$, the Lagrangian.
What exactly, was Lagrange's role, especially in the part that bears his name?
Although called Hamilton's Principle, there were a number of mathematical physicists of the day who contributed to the overall work. Contributions by mathematicians such as Euler, Lagrange, and similar or related work by Maupertius should not be ignored. Also, powerful add on work was done by Jacobi and Poisson. Plus, the original work with Calculus of Variations began (I think) with Johann Bernoulli's solution to the Brachistochrone Problem.
Also, I think that the pure Hamilton's Principle should be written as the stationary action: $$ \frac{\delta S}{\delta q_i(t)} = 0 $$ Where the action $S$ is written as you have in your post with the time integral of the Lagrangian.
