What term can be used to label the chemical change sodium cation undergoes after dissolving in water?

It seems that no chemical change can occur unless a new substance is formed, considering that the definition of chemical change is the transformation of "old" substances into "new" substances. This is an intriguing question.

However, since chemical equilibrium is a dynamic state, chemical conversions take place on a constant basis when a system of chemicals is in or near equilibrium.

Although all of the substances involved in these chemical conversions are already present in the system, they do not result in the creation of "new" substances. However, by altering the system's external constraints (such as temperature and pressure), a quantitative or net conversion (chemical reaction) is still possible.

If we want to refer to a single species (microscopic) or to the substance-macroscopic level, as well as to any change (qualitative or quantitative), perhaps the term "chemical change" is more general than "chemical reaction" in your question. In contrast, a chemical reaction is typically thought of from a phenomenological point of view (macroscopic change with observable evidence of new substance(s) formed).

To give a concrete example, let's take the example of table salt dissolving in water. We usually say that no "chemical reaction" occurs, meaning that no new substance is formed, but if you look closely, you will see that new species are formed, namely the hydrated ions. This change is best described as a chemical conversion, or a "chemical change."

Since the hydrated ions lack a unique and constant composition and cannot be separately isolated to appreciate any property due to their electrical charge, they are generally regarded as new species in this context rather than new substances, according to the IUPAC definition of "chemical substance."

I have two more examples here.

Acid ionization is universally understood to be a chemical reaction or change, but adding acid to water does not produce a "new substance"; rather, the acid is dissolved and becomes a water solution.

I wouldn't give it any term, because it isn't a chemical change at all. It would be labeled "no reaction", or "NR".

Chemical changes must result in formation of new substances, those that result from literal bond breaking and bond making, such that chemically-different substances are formed.

For instance, simply forming hydration spheres is not indicative of a chemical reaction; that is merely the motion of substances in a solvent to surround solutes and establish an energetic minimum. Furthermore, the former state does not exist for a practical amount of time.

That is,

is not a chemical reaction, because in this example, it is an ion-dipole interaction, an intermolecular force... and that does not make these substances behave in a chemically-different manner. But chemically different from what...?

The chemical transformation that sodium cation experiences after dissolving in water is referred to as "hydration."