Brown's family says that their son's civil rights were violated.
The family will announce the civil lawsuit on Thursday morning, attorneys said.
This is at least the third time that a memorial to Michael Brown in Ferguson, Mo., has been damaged.
With three council seats up for grabs, Ferguson grapples with its future.
The county police and state troopers would take over security for the protests in Ferguson.
In recent days, protests have broken out in Los Angeles, the Atlanta area, Wisconsin and Ferguson, Mo.
Eric Holder said Thursday he strongly condemned the shooting and praised steps the city has taken in recent days.
The county police chief said the gunshots did not come from protesters in the area.
Jackson resigned Wednesday, 214 days after Ferguson Police Officer Darren Wilson fatally shot teenager Michael Brown.
After holding out for months, the top cop in Ferguson resigns.