If you named your router "A" then changed it to "B", would they know / be notified
Does your ISP know your router's name
By Name I'm assuming you either mean the name of the WiFi SSID (wireless network name) or the hostname assigned to the router on your LAN.
In either case there is no need for the ISP to know this. There is no reason for either of them to be broadcast to the WAN side of the router.
However many ISP provided routers do gather large amounts of information about your settings. These are usually for diagnostic and usage monitoring purposes. I would imagine that for some ISPs everything the router can "see" can be read remotely - including both of these names. The ISPs argument for why would be for phone support - i.e. you've changed the router name to something different from what is printed on the back of the router and now you're phoning them asking why you can't find your network on your phone.
If you would rather they could not do this use your own custom router / access point.