As I think you've discovered, $HTMLExportFileName is generated from $Name unless you set a value. If you want to change the note title, one good idea is simply to manually set the note's $HTMLExportFileName to the existing automatically-generated value before changing the note title. This way the $Name changes but $HTMLExportFileName continues the same.
If you're just manually changing the $HTMLExportFileName, I'd stop and ask myself 'why?'. Who, apart from very few of us worry about the exact phrasing/spelling of web page filenames. Don't forget the impact on inbound links to your site. A seeming typo in a filename is a small embarrassment compared to breaking lots of links pointing at that page from elsewhere on the Web.
Anyway, so you've changed a few notes and now have old ones - what to do? I generally delete the local folder of exported files (i.e. sibling pages) but if your TBX is richly interlinked you need to think about the links in other notes that may now link to the wrong filename. Thus, given that Tinderbox export so quickly, I now tend to delete the content's of the export folder (except any elements not generated by Tinderbox) and re-export. That way all the intra-site links are now correct.
If exporting locally but using elsewhere, e.g. on your web-server on a different LAN, then I generally use FTP. Depending on the speed of access, I either repeat the process - delete the contents of the root folder and upload the new files, or I synch the two sets of files so the remote set (web server) reflects the local set.
- for things you're putting online for a while and which may be linked to by others, if you need to alter $Name, hard-set $HTMLExportFileName so it doesn't change.
- for local export, rather than find the odd old-named file, simply delete its siblings or delete the whole export and export the TBX afresh (it's quicker that hunting around!).
- if you're exporting page-by-page, set your TBX up so non-export pages don't export (by default everything does). After that re-exporting is a simple menu driven selection and a few seconds of export processing.
- if exporting public-facing pages, balance embarrassment over poor HTML filename choices over Web (inbound) link breakage if you alter a URL.