Its part of the 2 way Frequency Divider crossover.
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How exactly do you know that?
Just a guess that its a 2 way speaker. If its a 3 way it could be for one or the other speaker drivers.
Years of building, up grading or repairing 2 and 3 way speakers. Cross over networks are not rocket science, close, but not. To build one from scratch takes a lot of equipment I do not have. But, if the OP were to make his son pull the other speaker and take a picture that may help. The problem is the damage is just after the 0.35__. It is for a fact an Inductor and they are rated in mH so I must assume it is 0.35mH. It will also have its spec's silkscreened on it, like max volts. A picture of speaker 2's inductor may help.
I have taken so many "Top of the Line" audio equipment apart and replaced a lot of cheap components.
Most cross over networks will:
1. Match the speaker drivers Ohms so the net effect is 4 or 8 Ohms using resistors in series, parallel or series-parallel circuity.
2. Use Inductors and capacitors to change frequencies for each driver. It will have 'Taps' off for each set of drivers, Tweeters, Mid range woofers and Bass woofers. Now days even 'Sub' woofer drivers.
