I'm not sure if there's a word or phrase for a "reverse bibliography", but that's what I'm looking to put on my website - a list of other papers that have cited my own work. The goal is to show that my work has had influence and has spread beyond just the paper itself. Is this common in academia, or is it only common to list one's own papers?
For any well established research, they will usually have quite a large number of citations, which would make a "cited by" section would be exhausting to maintain and large beyond meaningful readability. A better route is to link to an database that actually engages in curating ones citations, such as Google Scholar.
I do, however, sometimes see people maintaining a collection of notable articles in non-scientific media. These are not normally picked up by citation databases and are valuable for showing that work is having impact outside of the scientific community, as well as (sometimes) providing a nice introduction to the work.