Is Echinacea suitable for a child of 11, who after having a cold develops a chronic dry cough that lasts for several weeks?

In principle, the answer is yes.

However, we should first check that the child has not developed a form of (temporary) asthma which may be manifested as a cough. This will be more likely if the cough is worse at night or with exercise (running around, playing in the garden etc.).

If you have not already done so, I would suggest that you speak to your doctor to ask if he could eliminate this possibility.