The sphere example you added at the end is not anywhere near something we should be applying a symplectic manifold conditions upon. The volume form you had (really, surface area form), is the correct one for seeing it as a Riemannian manifold, and it is ok that it is not of the standard canonical symplectic form because nobody should be taking it as a symplectic space. The problem we want solved, is finding how to get the appropriate momentum variables to those position variables, get the symplectic form, and Hamiltonian, all together at once.

@naturallyInconsistent I don't understand your objection at all. $(\mathrm S^2,\omega)$ is a symplectic manifold, and so you can do Hamiltonian mechanics on it - and indeed it is precisely what we should do if we want to obtain spin-$j$ systems via geometrical quantization. The "problem you want solved" is precisely what comes above that section of my answer - start from $Q$, construct $T^*Q$, and define the canonical symplectic form. You obviously can't get the Hamiltonian from that procedure because you have to specify the Hamiltonian yourself.

@naturallyInconsistent In your comment to Qmechanic's answer, you refer to figuring out the KE and PE, but one never simply "figures out" these things - they ultimately supply them. Conventionally we might make the Hamiltonian of the form $H(q,p) = p^2/2m + V(q)$, but that's not necessary (e.g. for electrodynamics). The "right Hamiltonian" is uncovered in the same way as the "right Lagrangian" - by figuring out which prescription reproduces the dynamics of the system you're trying to describe.

Firstly, if you admit that one would have to have a prescription to uncover the "right Hamiltonian", then you should also realise that, without such a prescription, you have not yet solved the OP's underlying question. That is literally the essence! Secondly, you can insist until your face is blue that $(S^2,\omega)$ is a symplectic manifold, but you will scarcely be able to convince people that your manifold that has no description of momentum is supposed to have anything to do with the Hamiltonian mechanics that people are interested about.

The fact of the matter is that we start with an initial (pseudo-)Riemannian manifold Q that we may lay down any convenient consistent coördinates, and we may construct $T^*Q$, thus getting tautological momenta, but that is not sufficient. We need some prescription for how to get the Hamiltonian and the identification of the tautological momenta with canonical momenta. If we can only ever start from guessing a Lagrangian or guessing from a huge pre-computed table, that is fine, but that should be stated clearly. If there is some prescription that omits guessing from Lagrangian, we wanna know!

@naturallyInconsistent I used the example of the 2-sphere to demonstrate how a symplectic manifold gives rise to Hamiltonian dynamics. If you wish to label my example as uninteresting (despite its connection to QM in finite dimensions), then that is your prerogative. The main point, however, is that there are two independent inputs to Hamiltonian mechanics - the phase space on which the dynamics plays out, and the Hamiltonian function which generates those dynamics. Given a configuration space $Q$, we can construct the phase space as $T^*Q$, but you continue to insist that [...]

@naturallyInconsistent [...] there should be some unambiguous way to determine a Hamiltonian from this data, which is absurd. Let $Q=\mathbb R$ so $T^*Q \simeq \mathbb R^2$ with natural coordinates $(q,p)$ and symplectic form $\mathrm dp\wedge \mathrm dq$ as described in my answer. Let $H_1=p^2/2m$ and $H_2= p^2/2m + m\omega_0^2q^2/2$. If you choose $H_1$, then you get the free particle, whereas if you choose $H_2$ you get the harmonic oscillator. You are free to choose literally any Hamiltonian you wish (though they may be physically uninteresting), so how could you possibly expect some [...]

@naturallyInconsistent [...] prescription to make this choice for you purely from the kinematical phase space (or configuration space, if that is your starting point)?

I particularly like your free particle v.s. harmonic oscillator example, and I agree that it would be absurd in the general case. But the whole point is to ask if a proper replacement of the old standard operating procedure of guessing that the Lagrangian is KE - PE, that we just have to express them in our nicely chosen coördinates, and Legendre transform to momenta and Hamiltonian, exists or not. If it does not exist, then nobody should be claiming that this problem is actually already solved, i.e. assert that we could jump straight to Hamiltonians and momenta on a generic symplectic space

@naturallyInconsistent The replacement for guessing a Lagrangian and performing a Legendre transform to obtain a Hamiltonian is simply guessing the Hamiltonian from the start. Of course, to the extent that the Lagrangian isn't a blind guess (e.g. you could impose symmetry constraints) then you can reason in analogous ways about the Hamiltonian. But there's no magic here - you need to provide L in Lagrangian mechanics, or H in Hamiltonian mechanics.

Sometimes you can get one from the other. But at some point, you yourself need to prescribe the dynamics of the system you're trying to model.

The original question was, in effect, how to get the "right" canonical momenta given some position coordinates. The answer is essentially trivial - if you construct the cotangent bundle, a set of position coordinates naturally induces a coordinate system on the contangent bundle, and those are your q's and p's. Somehow the question of how to write down a Hamiltonian got mixed in, and the answer to that is similarly trivial - literally write down any smooth function of the q's and p's.

Of course, a randomly chosen function won't give you the dynamics you're looking for, so then we need to be more precise. Experience with systems which have equivalent Newtonian or Lagrangian formulations tells us that often - but not always - we can write H = T + V where T is quadratic in the p's and V depends only on the q's. You can impose the requirements of various symmetries on the system, you could add perturbations or variations to previously understood models, and the list goes on.

Given a generic symplectic space with no a priori physical interpretation for the coordinates, it is indeed not obvious how to pick a Hamiltonian. My example with the 2-sphere was, in part, meant to illustrate that fact. But it's not a problem which has a fixed solution - you have the freedom to construct any model you like, and deciding on the most appropriate one for your situation is why physicists get the big bucks (/s)