Changing the 'hover name' of a url /destination in InDesign

It can be done with workarounds, as Bob notes, but the purpose of this pop-up is to show the actual link contents, for security and online safety reasons. Most browsers etc. are hard-coded to present the actual destination link, no mattrer how concealed or obfuscated it might be, to fulfill that important need.

While having a "message" pop up instead might be nice for a document or marketing reason, if it were possible 'bad actors' would use it to conceal a malware link. So there are no simple, "good" ways to accomplish this, and most of the reasons lie with the browsers, not the code or documents.

Convert the image to a button and add the description in the buttons and forms panel. And don't expect any of this to work in a mobile device or any third party PDF reader if your intent is PDF.