I need to move my router to another room, but the cable going from the splitter the cable company installed isn't long enough to reach the new room. Can I simply buy any coax cable and use it in place of the one the cable company used? Is there something special about it? It doesn't look like it, but I'm betting that if I ask them, they are just going to tell me that they need to send somebody out to replace it and charge me $70+ to send out a technician.

Yep you'll just need some RG6 coaxial cable to hook up to the modem. If it is specifically the router you are wanting to move you will need Cat5/6 Ethernet cable.