I have not heard the term dry tonic before. What is a dry tonic?

It means tonic water, or Indian Tonic Water. It originally started out as a medicine. British medical doctors realized quinine powder would help prevent malaria, and it could be mixed into water for drinking. The problem was, it was so stinking bitter nobody wanted to take it. The solution was to mix it with gin, which the soldiers and officers would drink. Over time, they developed a taste for the bitter water. Modern tonics don't contain quite as much quinine, and a lot more sugar and other flavorings as well. I'm not sure where the "dry" bit comes in, but I imagine it has to do with the balance of quinine and sweetener. Quinine itself is rather astringent, which leaves your mouth feeling dry- think of the red wine that needs to stay in the bottle a bit longer, and how it leaves your mouth feeling. More quinine in the water makes it feel dryer in the mouth, and ups the bitter factor as well. More sugar would make it easier to drink. Given that suze itself is bitter, I'm not sure how that would be an appealing combo.

Nope, not mineral water or club soda (though you could use those in a pinch and they would give a different taste).
Tonic originally and usually means a carbonated drink which includes quinine.
Most now also have sugar.
I'm guessing "dry tonic" mean one lower in amount of sugar used.