<t>The UI does not directly support renaming a Function, but you can work around this using the following manual steps:<br/>
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- Stop your Function App. To do this, go under Function app settings / Go To App Service Settings, and click on the Stop button.<br/>
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- Go to Kudu Console: Function app settings / Go to Kudu (article about that)<br/>
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- In Kudu Console, go to D:\home\site\wwwroot and rename the Function folder to the new name<br/>
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- Now go to D:\home\data\Functions\secrets and rename [oldname].json to [newname].json<br/>
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- Then go to D:\home\data\Functions\sampledata and rename [oldname].dat to [newname].dat<br/>
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- Start your function app, in the same place where you stopped it above<br/>
In the Functions UI, click the refresh button in the top left corner, and your renamed function should appear<br/>
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Note: doing this can lose some historical logging.<br/>
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Github Issue for renaming Azure Function<br/>
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Edit for new info<br/>
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To anyone like myself that arrived here looking to rename their function, despite this being the previously correct answer, there is now a much smoother CMD based process as detailed in this answer by SLdragon and an even smoother GUI based process detailed in this answer by Amerdeep below.</t>