For each site, you can hard-code different search button label.
Perform the following:
EXAMPLE: For more information about the contents of the file, see the CustomButton.ascx.cs in Sitefinity SDK: documentation-samples on GitHub.
Build your solution and restart the application.
When you place the standard Search widget on a page from FirstSite, it displayes label Search; when you place it on a page from SecondSite, it displayes label Search2; when you place it on ThirdSite, it displayes label Search3.
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