Maybe you get lucky and they used ethernet wire to do the telephone port and you can just change out both ends to ethenet.
I would consider buying another router and use it as a AP. You would then place your current router near the ONT and place put the new router running as a AP in the dining room to provide the wifi.
Now if you really want to learn some stuff you can buy a couple small "smart" switches that support vlans. You would put one in the room by the ONT and a second in the dining room.
These are still pretty cheap, tplink 8 port ones are about $25. What you can then do is carry 2 different vlans over the same cable. In effect you make 1 cable into many. Not hard but it takes a while before the concept of virtual things becomes clear.
The downside is you will cap your internet bandwidth to 500mbps because there still is only 1 actual gigabit cable and you are sending the data over it twice.