Some modifiers have an option to affect only specific parts of a model defined by a Vertex Group.
The workflow is like this:
In Edit Mode select the vertices you want to affect, create a vertex group by pressing the Plus Sign on the right of Vertex groups.
Click then on Assign to make the selected vertices part of the newly created group.
To use the modifier on a specific vertex group, select it from the list available groups:
Vertices that are not part of that vertex group will not be affected by the modifier.
NOTE: If what you are missing are faces on the mesh, the solidify modifier will not fill those gaps. It will only make the surfaces of existing faces thicker.