Diagnosis
The side burner has a separate thermocouple safety valve. If it doesn't sense a flame heating it up within a few seconds, it shuts the gas off. The flame might not be hitting the thermocouple tip, or the thermocouple itself is bad.
How to Fix It
Locate the thermocouple. It's a small copper rod with a metal tip that should sit directly in the pilot flame or the main burner flame. Make sure it hasn't bent away from the flame path. While holding the control knob in, light the burner. Keep holding it for a good 30-45 seconds after lighting to let the thermocouple get hot enough to signal the valve to stay open.
If That Didn't Work
The thermocouple might be faulty. You can test it (carefully) by holding a lighter flame directly to its tip for 60 seconds while holding the knob in. If it stays lit after you remove the lighter, the burner's flame alignment is off. If it still goes out, the thermocouple needs replacement.
Heads Up
The side burner is more susceptible to wind blowing the flame away from the thermocouple. Try to use it in a sheltered spot or on a calm day. Also, spiders love these tubes too—check for webs clogging the burner ports.