Rube Goldberg Science Project
The string needs to have a very low melting temperature. The wire would have to be much much smaller than  human hair for 9V to get 10 watts into 8 Ohms in a short strand. (0.008mm)  But that hair would be the ideal plastic string for a fast break.

190–350 °C is common for nylon.  

A 9V battery may not be an excellent choice

A 10 Ohm resistor might  over 160'C but then the epoxy starts to smoke (toxic).  

A tiny lightbulb filament but it has to get hot enough to burn a plastic thread then the plastic starts smoking on the glass. (toxic)  

Brown garden hemp string may work but the light bulb needs more power and it might take a minute or so unless you have a very thin filament.

If you can find a 9V high power flashlight, it "might be borderline hot to melt" with some fire-proof insulation underneath.



But most likely unless it can melt in boiling water, it wont work.

added

When my kids were small we used a 12V lattern battery to power an igniter (heater wire) for starting a solid rocket. A Hobby shop will carry these small glow igniter wires (  which if you have a momentary switch to use it wrapped around a wooden match, that will work. Safety of mounting the match is important. Mount it inverted so the flame doesn't burn the wire as much and falls into a bucket of water.