What makes a good two-player paper game?
The best ones start almost instantly. You should not need a long explanation, special equipment or an account. Tic-Tac-Toe works for a two-minute gap. Dots & Boxes and Four in a Row give you more strategy. Hidden Fleet is better when you have ten or fifteen minutes. Nim and Sim are especially good when you want something that feels simple but rewards thinking.
Online or on paper?
Both. The online versions are useful when you only have one device. The printable library is there for journeys, classrooms, restaurants and anywhere you would rather put the screen away.