I second Aeryk's suggestion to focus on low-dimensional topology. More generally, I think that algebraic topology can be more exciting than point-set topology.
That said, when I was an undergraduate, I do remember being quite excited about the Bourbaki program and point-set definitions and so on. I remember one "first course in topology" that alternated days: low-dimensional topology on even days and point-set on odd. Except the point-set portion began with set theory, cardinal and ordinal numbers, and the axiom of choice; then moved on to metric spaces; and only then introduced point-set topology. I basically think that to motivate the point-set definitions, you had better start with metric spaces.
If on the other hand you focus more on broadly-defined algebraic topology, then in addition to the low-dimensional topology of manifolds (surfaces, knots, etc.), another good topic is Brower fixed-point theorem as an application of fundamental group functor on pointed spaces. Perhaps, if you are very ambitious, you can prove that 2dTQFT = commmutative Frobenius algebra, and talk more generally about cobordism equivalence. Oh, and especially given the recent sad news, be sure to include a little Morse theory and Outside In.