Recently purchased Sony Bravia KD49XG9005 ethernet connected to my DSL router and a Win 10 NUC where I have my Plex Media Server. Not always but frequently I'll select the Plex app but the server isn't found. After I retry two or three times it seems to find it. The NUC is always on but maybe it's a setting somewhere? I reconnected everything here in the UK after moving from the US where it was connected to an old Samsung TV. It also doesn't connect to iPlayer and Netflix once in a while.
We have pretty poor internet and I'm having to reboot the router every couple of days. Could that be the cause?
It could be your Internet. Plex relies on the internet connection to authenticate your connection, even when connecting to a local Plex sever.
Take a look in the Plex settings, select "Network", select "Show Advanced", and look for the "List of IP addresses and networks that are allowed without auth" box (near the bottom). You can either enter a single IP address, or a range and mask, so something like "192.168.1/255.255.255.0" to allow any device on your network, if you are using the 192.168.1.x range.
It's not something I've had to do myself, so I've not got personal experance of it, but I have seen other people having issues with authenticating with no Internet, so maybe that'll help you (assuming your connection issues are because of poor Internet conectivity!).