To sum up: it's up to you to choose what an interesting effect size is.
There are several measures of effect size, not just the variant of Cramér's V you mention. Here is a non-exhaustive list*:
- Cohen's w,
- Fei coefficient,
- Pearson contingency coefficient
(see the effectsize R library for more information about these last two).
These coefficients can be converted between them, and in particular to Cohen's w. Cohen's w is often used for power/sample size calculation (e.g. see the pwr library in R).
You may stumble on some websites mentioning some general guidelines for the interpretation of Cohen's w or Cramér's V. These guidelines generally come from Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed) (see this question for more information about that).
But what is an interesting effect size is ultimately up to you, there are no universal guidelines that apply to any situation. As Cohen says in his book (p.224):
The best guide here, as always, is the development of some sense of magnitude ad hoc, for a particular problem or a particular field.
In other words, the question you have to ask yourself is, for example: does it matter in any way, if some cells deviate of 1% from their expected values? Would the following table be interesting for your research question?
| state | state A | state B | state C |
|---|---|---|---|
| percentage | 0.35 | 0.33 | 0.32 |
If it's not interesting, what about this one:
| state | state A | state B | state C |
|---|---|---|---|
| percentage | 0.37 | 0.32 | 0.31 |
or this one:
| state | state A | state B | state C |
|---|---|---|---|
| percentage | 0.45 | 0.3 | 0.25 |
Each of these tables has a different effect size. Read about previous research on your ongoing subject of study, think about what kind of tables would be little interesting, somehow interesting, very interesting, with a difference between cells so large that it would be completely unexpected, or with a difference so small that it would not matter the least.
Then calculate the effect size of these hypothetical tables. And there you go, you know what effect sizes you can qualify as "small", "medium", "large", "astronomical", or whatever qualifier you'd be happy to use.
Preferably, you should do these calculations before running your study, as sample size calculations require you to pre-specify the smallest effect size you're interested in. Maybe add some safety margin to your initial calculations to avoid ending up with an underpowered study.
*NB: Unless I completely misinterpreted it, researchers in genomics seem to use different measures of effect size for power/sample size calculation than those I mention above (see https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4133582/ for example), probably because they deal with complex categories that come in millions or billions. But that's not my field at all, so I can't say much about that, and I may be a bit off the mark.