Solution: the development diagram of the cylinder is as shown in the figure: the shortest route from A to B with a cotton thread along the edge of the cylinder is: AC → CD → DB;

That is to say, in the unfolded rectangle of the cylinder, the rectangle is divided into three small rectangles on average, and the route from A to B along the diagonal of the three rectangles is the shortest;

The radius of the bottom surface of the cylinder is 2πcm,

∴ The width of a rectangle is the circumference of the bottom of the cylinder: 2π×2π= 4cm；;

In addition, the height of the cylinder is 9 cm.

∴ One side of the small rectangle is 3 cm long;

According to Pythagorean theorem, AC = CD = DB = 5cm；;

∴ac+cd+db= 15cm；

So choose C.