Does it matter what the input voltage is to a DC/DC step-up converter?
If your source (batteries?) can deliver the required power, it doesn't matter what the input voltage is as long as it's in the range.  The current draw from the batteries will be higher as the voltage gets lower for a constant output load.

A DC-DC converter with a constant output load is a constant power device.  In your case your converter puts out 5W (5V*1A).  There are some conversion losses, so the input power will be higher, for example maybe 6W (the output power divided by the efficiency.)

So your input must supply 6 watts.  At 6 volts that will be 1A.  At 3V that will be 2A, etc.