Are you using a Sony TV? Does the message say something like "...press the HOME button for more information..."?
Apple
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I have the same issue. I get a „lost internet connection” message every now and then. After some research (google&reddit 😋) the issue for me is that I’m surrounded by WiFi. WiFi routers, direct connect tvs, printers, even some soundbars have WiFi, smart home accessories. Even though most of these networks are hidden they cause interference. You can try to change the WiFi channels on your WiFi access point but if it’s like mine it won’t help. My solution was to get an WiFi extender (don’t get cheap ones they mostly help over a very small distance).
I suggest you to troubleshoot and if possible move the Apple TV closer to the WiFi access point.
The money you spend on a new ATV with an Ethernet port will be tiny vs the value of your time and stress trying to fix wifi dropout issues. In my experience ATVs are quite stable over wifi, so my best guess is your older 1st gen’s WiFi adapter is sick. You could also try changing the 5G wifi channel in your router settings, in case a neighbor or another device is competing with it.
Reserve and IP address for your appleTV on your router.