Physical origin of the inflaton field?

As @Rennie states, no-one knows what the inflaton is. The current state of affairs is that it is generally accepted that a period of exponential expansion took place during the early universe. This explains many of the features of the observable universe that are otherwise extremely hard to explain. One of the big industries in cosmology is to try to build a sensible and well motivated model of inflation that agrees with the experimental data. To date there are hundreds of such models, but many of them share a common feature which is that the inflation is produced by the potential of a scalar field. When this is the case, the particular scalar field in the model is known as the inflaton. Examples of models include:

1. Higgs inflation where the standard model higgs boson plays the role of the inflaton. Since the Higgs is the only fundamental scalar to have been observed so far, it is an important question whther it could be the inflaton.
2. GUT inflatons. In grand unified theories, there are a lot of extra scalar fields which must be present to break the GUT symmetries, these could play the role of the inflaton.
3. SUSY inflation. In Supersymmetric models, scalar fields abound and many extensions of the MSSM require additional scalar fields, any of these could play the role of the inflaton.

There are many many more models, all have various pros and cons. The important point is whether the scalar field in the model has a potential that could produce inflation and how the predictions from the specific model agree with experiment. As CMB data gets more refined, some models will be ruled out but unambiguously identifying which scalar field in nature is the inflaton is a long long way off.

The potential of the inflaton field has to have a particular form to produce inflation. Inflation requires a negative-pressure vacuum energy density. This is generically produced by a scalar field as long as $\dot\phi^2 < V(\phi)$. So the requirement boils down to having the scalar field sit at a point where the potential takes a large value and is not too steep. This can either be a local minimum (false vacuum) and the field eventually tunnels out, thus ending inflation or a flat potential where the field slowly rolls down the potential. Slow roll inflation is now generally preferred since false vacuum inflation has problems with reheating.

Remember, at the moment, producing models of inflation is very much in the realm of model building and there are still models that do not even invoke an inflaton field.

The source of inflation is the inflaton, but no-one knows what the inflaton is!

The inflaton potential is calculated by looking at the universe and fiddling with the potential to get something that fits observations.

In other words, there are no fundamental theories that predict the inflaton properties. At the moment the theory is purely phenomenological. However this does not mean it lacks predictive power, as we get more information out of the theory than we have to put in as parameters. Hopefully the Planck satellite will give us more information about the theory.