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Could you explain where in the ticket, the person's name is selected in the process?
Are they assigned to the issue in later status? If they are assignee , you can leverage the notification scheme - assignee role to be notified.
You can find more details about the notification scheme in https://support.atlassian.com/jira-cloud-administration/docs/configure-notification-schemes/
If that is the case, their name will be selected from a user picker custom field. Is that right?
Refer to https://support.atlassian.com/jira-cloud-administration/docs/configure-notification-schemes/ again , and configure to notify that 'User custom field value' value for that.
If you are the jira admin, you should be able to configure the notification scheme
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