You should install screen
opkg update
opkg install screen

Connect to Omega with ssh and run screen:
screen /dev/ttyS1 9600
Now you should see the output of those Serial.print on this Arduino IDE Serial Monitor like "screen".

To kill this screen session, type Ctrl-a then k.

See also: Onion Omega2 Documentation Communicating with Serial Devices
Using the screen Command

Good luck!