Rebuilt laptop: new client, but retain old backups..?

Mark any backups you absolutely want to keep as “archive”, then download and install the client for that PC from the web interface.
I would then tell it to do a full backup (image and files if you do both) so that it’s got a clean base to go off. It will probably do that automatically anyway, but just to be sure.

If the backups are marked as archives, it then won’t delete them if it decides the new ones are going to take too much space. It’ll just stop and you can decide what to do later.