Mechanism-led reasoning: Theophylline has two big actions, and the trick is matching the right one to diuresis.
PDE inhibition raises cAMP and drives bronchodilation, CNS stimulation, and positive cardiac effects, not the renal sodium handling. So options for PDE3 and PDE4 inhibition do not explain the diuresis.
Beta-2 agonism is not a methylxanthine action at all, so that option is out.
The diuretic effect is a renal tubular event. Methylxanthines block adenosine A1 receptors on the proximal tubule, which normally promote sodium and water reabsorption. Antagonising A1AR cuts proximal reabsorption, and the effect disappears entirely in A1 receptor knockout mice. Theophylline also weakly dilates renal vessels and boosts renin, both via adenosine receptor blockade.
Pearl: adenosine = retain sodium; block it (theophylline, caffeine) = mild diuresis.
Ref: renal pharmacology of methylxanthines.