Neil,
As Dan says, the entry hole at the very top of this games' subway system is shielded by a single drop target. If the option chosen when launching a ball into play requires entry to the subway, a small coil (more a relay coil rather than a solenoid coil, really) operates a linkage or 'finger' which pushes on the target in the same way as a ball hitting it. Then the pull-down spring retracts the target below the playfield. That's the idea, anyway. A common problem with Star Trek Next Gen, seemingly more so than other games with this arrangement, is a partial breakage of the holder for the dropper coil. This allows the coils' metal plate, with the finger attached, to move a long way away from the coil - this also affects the distance of the finger from the target (it should be quite close). It's my impression that this can lead to the dropper coil being overworked, resulting in failure.
Have a look at the dropper mechanism, which is attached to the target frame. Locate the small coil, and its metal plate. There should be a long curved section extending from the dropper coils' bracket, to provide a stop for the plate (and to prevent it from falling off). If the only thing securing the plate is its return spring, then yours is another example of this problem. It's not all that difficult to correct though. Obtain a "Frame & Eyelet", and the coil, plate, finger and spring can be transferred from the existing one without soldering. Just take care to fit the plate facing the right way, and when bending the central tab to retain the plate/adjust the finger.
