Looking for a statistical term close to "precision"

I don't think there is perfect standard terminology, but I think we can reason our way to a viable option.

Let us imagine that we are using a person's height as a feature for a classification problem, like predicting health outcomes or athletic performance. How do we expect the model to change if we are able to measure height accurately to within a millimeter rather than just an inch? It is natural to expect the model accuracy to improve, but this could depend on the details of how the model is fit to data - using millimeters could cause it to overfit, for instance.

I would argue that the direct, first-order effect of using millimeters instead of inches is that the model will likely become more discriminative: the probabilities that it assigns to different classes will be more concentrated in a small number of classes with the more refined units. So it makes sense to me to apply this label to the units of measurement and say that millimeters are more "discriminative" than inches. If we need a noun like "precision" then I guess it would be "discriminativity", though I'm not sure that's actually a word.

This choice of terminology has a number of nice properties. It has a generally positive connotation (discriminative models are usually desirable), though the statistically literate reader will be cautious about overfitting and wonder how much of an effect making certain measurements more discriminative will have on the statistical power of the overall study. This is probably how the reader should approach a study which measures a person's height in millimeters rather than inches, so it fits your example well, at least.

I have heard statisticians and data scientists use the word 'granularity' to express the idea you are looking for. Here is a quote from the dedicated Wikipedia article: 'The granularity of data refers to the size in which data fields are sub-divided'. The article also gives an enlightening example of proper usage: 'A kilometer broken into centimeters has finer granularity than a kilometer broken into meters'. Here is the link to the article: https://en.wikipedia.org/wiki/Granularity#Data_granularity.