I have a Hamiltonian function, $H(p,q)$ which is a function of $p$ and $q$: $$H(p,q) = p^2 + q^2$$
and I also have a Lagrangian via the Legendre transformation which is
$$L(q,\dot q) = p \dot q - H(p,q) = p \dot q - p^2 - q^2$$
I want to find the costate $p$, which is a function like this: $$ p = \frac{\partial L}{\partial \dot q}$$
From the Hamiltonian dynamics, I know that $$\dot q = \frac{\partial H(p,q)}{\partial p} = 2p$$
So, in this case, what is $p$, since it is a partial derivative of the $L(q,\dot q)$?
EDIT: Is it $p =\frac{\partial L}{\partial \dot q}= \frac{1}{2}[\dot q - 2 p]$ ?
The notation here is slightly confusing. $L$ is really just a function of three a priori unrelated variables. Consequently $\frac{\partial L}{\partial \dot{q}}$ is just the partial derivative with respect to the third argument of $L$, which in your case is just $p$. The fact that the third argument is the time derivative of the second argument doesn't come into play at this stage.