I'm looking for a way for a Pro project to "remember" certain settings, such as what value was selected from a ComboBox. Does a project have local storage to save and retrieve custom configuration settings?
You can read and write the Project's settings using these two overrides on the module class -
OnWriteSettingsAsync and OnReadSettingAsync.
There is a sample in the arcgis-pro-sdk-community-samples repository that demonstrates this.