Generally speaking, what does a "slow-growing" tumor mean? for example, how much larger would a slow-growing 1mm tumor be expected to grow in a year?

This depends on many factors, like what type of tumor is it, where is it located? Can it be treated to slow the growth further? Without knowing more details, it is not possible to give a specific answer. You also need to know if it is benign or malignant. Slow-growing is better than fast-growing. Best of luck to you!

Carcinoid & prostate tumors are slow growing and might be expected to double volume every 6 months. Certain lung tumors are known to be slow growing. In past years, doctors considered prostate tumors to be so slow growing that it became a fad to ignore them and remind patients that “something else would kill them before the prostate cancer”. This was about 20 years ago, and idea is now foolish.

Tumor growth is measured by the time it takes cells to double in number. Doubling the number of cells does not increase the VOLUME of a sphere ( assuming the tumor is relatively spherical ) very much until the sphere becomes pretty large. Doubling times are generally in 3 to 9 month range. So a 1 mm tumor might be 1.3 mm in a year. If tumor is in a bad location that increase could be serious.