Just tried it again... It remembers, as long as the number I enter doesn't have more decimal places than the set resolution of the configuration.
For example, if I set it to 0.00001, it forgets after 2 uses. If I enter 0.0001, it remembers forever.
I assume this is because it rounds? So, it has a (5 or more decimal place) non-zero value when it's closed the first time, which it remembers when the dialog is brought back up. However, this second time, the number is rounded to 4 decimal places, making it zero. Which is an acceptable entry.
But because it doesn't remember a zero, the next time it's brought up, that box is greyed out again.
EDIT: Pretty sure this is the case. Changed the display format to 5 decimals, and it remembered 0.00001. Same when I changed it to 3 decimals... It remembered 0.001. The text box in the Tool Height dialog defaults, and rounds, to the display format.