[MurMan]

This idea assumes that the interior of the tower is empty space that could be used by the moving weight. Towers have an elevator for maintenance personnel that would reduce the available space. I'm guessing that the towers aren't high enough to store enough energy to justify the cost of the extra equipment.

I doubt that suspending the weight is the main structural issue. It's the high center of mass when the weight is at the top.

[mNovak]

Yup, internal space is a very valid issue, given that cheap weights are probably big. I'd worry about oscillations for a high-up, suspended weight too.

Regarding justifying the cost--obviously depends what the cost is. If it's really just the generator, cabling etc, it might work as a short term load balancer. I saw somewhere about GE investing $X millions to smooth turbine output over 5min intervals.