Supplying steady 5v to the board from inconsistent battery pack

If you overload a supply the voltage will start to fall. The reason the Espruino resets is that the in-rush current for the relay coil is pulling down your supply voltage.
There are usually datasheets for relays with this information on.

@lavamantis
I would suggest working back from what you are switching.

• What device on the relay are you switching? i.e. boiler, heater, light etc.
  
• The voltage and current are you switching with the relay? i.e. the device load.
  
• How long are you switching the device on for?
  
  

The ULN2003 will switch up to 500mA. The relay you are using has contacts rated at something like 10A @ given voltages. As you can see this is a big difference.

Bi-stable relays or 'flip flop' relays switch the contact state each time you energise the coil. Again it depends what you are switching. If the power fails you need to make sure it fails to a safe state!