What CMS or hosting platform is being used? Most CMS's allow for "search" to be added to the site (plug-in or add-on). You can add it to the site and then remove it after you're done checking for the old brand name.
Alternatively and depending on your comfort with HTML, use a web editor, specify the URL and download all pages to the folder you specify on your hard-drive. If not familiar with web editors, do a web search on this term or contact a web developer (many not all web developers have one or more web editors for their use). For web editors I've used in the past, it's File / Open or something similar and then you get the option to "import" the contents. Once it's downloaded, most web editors will have a Find/Replace. Use this tool to locate all instances of the old brand name.
Alan