It is not possible to rename the channel integrations in Linnworks. 

The channel name is used as the Subsource for all orders downloaded from that channel so you cannot edit this. All channel integrations have a set naming method, i.e. eBay channels are always EBAY followed by a number, which depends on the order they were integrated, with the first eBay you integrate being EBAY0. Website integrations, such as Magento, use the URL of the site as the channel name, and Amazon channels use the Merchant name entered during the integration as the channel name.