Extension of cable AWG 24 ~30cm length with dupont connectors.
Powered with ~6.5V and enough available current (up to 2A).
Current extension is already AWG 24 ~50cm.
So ca. 1m is where the motor just silently "clicks" (ca. 0.5sec - 1sec intervals).
Could the amount of connections between cables make the voltage drop or something which causes it to stop working?
I figured using relatively thick cables (AWG24 instead of typical AWG26+) would make up for that.
Connections of any kind in a path degrade signals, or any electric flow. You can actually "see" solder joints with special equipment (TDR - time domain reflectometer).
DuPont female connectors seem to have less-than-good continuity. Try soldering or screw terminals.
I expect you will lose over 0.5V just in the wire at 2A. There will probably be at least another 0.75V at the connectors. Now when the motor starts it draws a lot more current so there will be additional voltage drop. Next thing to consider is your power supply impedance, it is not zero ohms so you will get a drop there.
@xfpd gave you some good advice and also consider a larger gauge wire. What is your power supply rating?