You can have an automation run when the multi-select contains that option, combined with an editable date/time field that has permissions set such that only automations are allowed to update timestamp field.
When the automation runs, update the record with the timestamp. You can get the timestamp from a script. If you are not comfortable writing that script yourself, I include one in my package of Automation Helper Scripts that I sell.
Note that this system has the potential to use up a lot of automation runs.
Trigger: When record meets conditions.
Conditions: multi-select contains the chosen option and the timestamp field is blank.
Action 1 (run script): run the automation helper script that outputs the timestamp.
Action 2 (update record): update the triggering record to set the timestamp field with the value outputted by the script.