How do I check whether my system supports network booting?
I don't see an ethernet option in my boot priority list.
You will not necessarily see the name of your network adapter in the boot priority list. You may see "Network Adapter" or "PXE Boot" instead.
To get an exact answer I would consult the manual for your motherboard.
My mistake. There LAN option was disabled in another section. Now it is working.
On one machine I enabled LAN Option ROM
under Advanced > Onboard Devices Configuration. On another, a similar option was available under a menu item named Northbridge (or was it South bridge? not sure).