You're not able to fully delete patient records, clicking the red "X" just puts them in a 'Disabled' status which keeps them from showing up in any of the search/auto complete fields.
If you are on the manage patients page, you can change the tab at the top right to view Disabled, Inactive, or All Statuses and you can find the old record and modify it accordingly.
Click the green 'Re-enable Patient' icon if you wish to make the patient record 'Active' again if it's status was 'Disabled' or 'Inactive' (see screen shot).
Click the blue edit pencil icon if you wish to edit or change the patient record.