Certain aspects and embodiments of the present invention are directed a sand control device cleaning system that can be disposed in a wellbore through a fluid-producing formation. The sand control device cleaning system can include a housing and a fluid communication structure. The housing can be coupled to a section of a tubing string of a well system. A portion of the housing or a container disposed within the housing can store a dissolving material. A dissolving fluid that can dissolve or otherwise remove particulate material can be formed from the dissolving material. The fluid communication structure can communicate the dissolving fluid from the housing to a sand control device coupled to the section of the tubing string.