Below "Edit Channel Header" add a separator line and then the option "Close Channel". See the "Leave Channel" option in public private channels as an example of styling.
Closing the channel should remove it from your sidebar (similar to clicking the "x" when hovering on the channel in your sidebar)
Note: Mobile already has this implemented.
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@mattermod I'll take a bite at it.
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PR: https://github.com/mattermost/mattermost-webapp/pull/3694