Noisy mattress coils are a sure sign that your mattress has worn out. The coils do not have their original stability and strength and the mattress should be replaced for proper comfort.
The type of coil can affect how noisy it is. Offset coils are hourglass shape and are hinged at the top and bottom of each coil. Less expensive Bonnell coils are more common and are knotted at the top and bottom, which creates more noise.
Check the warranty of your mattress; many cover squeaky coils if it is still valid. If so, contact the store where you purchased the bed to return it.
Flip the mattress over if it is two-sided to change the pressure put on the coils. Place a piece of plywood between the mattress and box spring to add pressure to the coils. This may minimize or eradicate the noise.