Park swings... at least the kind you sit on... swing on a horizontal axis that is parallel to the ground. So, you want to hang your sign vertically from a horizontal rod (axis) and make it swing to and fro plus and minus twenty degrees or so? Sort of the way a pet-entrance door swings up and down to allow a small cat or dog to enter and exit the premises? Not like a people-door which swings to and fro on a vertical hinged axis, right?
The easiest way I can think of to do what you want is to direct a small fan at one side of the sign. The breeze from the fan will push the sign away through a short arc, and when it reaches a certain angle the breeze will not be sufficient to support it there, so the sign will swing back down until it catches the breeze again. The weight of the sign (although small) will be sufficient to make the sign act like a pendulum. If the fan breeze is precisely directed, you should be able to use it to "push and pump" the swinging sign, much as your would push and pump a child on a playground swing. If you want to get really fancy (and more reliable), you could glue a magnet near the top of the sign that would actuate a small magnetic reed switch mounted on the axle rod at just the right position in the swing motion to turn on the fan on to provide each "push." Some experimenting will be necessary to discover how much breeze you need and how it should be directed at the sign. A small 12 V DC fan, like the kind used with home computers, will probably be sufficient. I would not want to mount the sign outdoors where Nature's breezes can easily overwhelm your purposeful breeze.
