Minimum wire gauge for 5 Volt, 5 Amp system
About AWG 20 (0.812mm) should be acceptable, use wire that is rated for 90°C or better.  That's much heavier than typical telephone wires, and higher temperature rating than some. If you need the OD of the wires to be small or if you anticipate high ambient temperatures, you can use PTFE-insulated wire, which is often rated for 200°C. 

I based that on a voltage drop of 0.5V (which will result in a change in brightness of about 25% with a white LED) and average (round trip) length of 3m, so resistance should be &lt; 30m\$\Omega\$ per meter. It's also the NFPA ampacity recommendation for AWG 20.