That would change the commit hash, at least if you want it to survive a clone of the repo. Of you'd store it externally so that it would only be able to be shown in the webui then it's of limited use, but maybe better than nothing.
(Not sure if such a field already exists in the commit blob)
That's not how git works.
This design doesn’t preclude the usage of additional information in the commit blob that isn’t used to compute the hash.
(Think for example how file access times do not affect its hash)
https://git-scm.com/docs/git-notes
"Adds, removes, or reads notes attached to objects, without touching the objects themselves."