At a high level, to rename a tag in Day One, you need to do the following:
1. Add a new tag to the entry/entries.
2. Remove the old tag from the entry/entries.
If you need further assistance on how to do the above, follow these steps:
Open the Journal Drawer and tap Filter, tap Tags, then tap the tag you wish to rename. This will apply a filter to your Timeline and only show the tagged entries.
Use the Multi-select Tool by swiping on an entry in the Timeline and tap the checkmark icon.
Tap Select All.
Tap the tag icon (near Select All), type your new tag, then tap Done.
Tap the tag icon again, then type your new tag again. (You should see it appear in the list of tags with zero entries with that tag.) Tap the tag to apply it to your entries.
Tap the delete icon (red x) next to the tag(s) you wish to remove from the entries.
Tap Remove in the confirmation alert.
After you've finished removing the tag(s) you don't want for the selected entries, tap Done.
In the Timeline view with All Entries selected, click the filter button icon > Tags > then select the tag you wish to rename. This will filter your entries in the Timeline.
Select all the entries for this tag by clicking the first entry, then pushing CMD-A on your keyboard or Select All in the Timeline filter at the top.
Control-click a selected entry to display the contextual menu, then select Tag....
In the tag-editing menu, type your new tag in the Search/Add field.
Delete the tag(s) you don't want by clicking the "X" button.