At the risk of introducing thread-creep (why we sing) and cliche early in the discussion, it depends on what you want to achieve with your song(s). (Amuse? entertain? provoke?)

one:
Start by singing songs you really like, changing the words and tune to suit your current performance skills and experience, don't bother about emulating others, do your best with what you've got. In any halfway decent singaround you'll be respected for having got up and performed irrespective of skill.

two:
Keep going to the singaround, performance improves with practice (less nervous etc.) plus the better your rapport with the other singers the more support you'll get during your songs (which can really help help build confidence).

three:
Don't be afraid to do songs the group doesn't know but generally keep away from songs someone else in the group already performs.

All very general, all very obvious, but without knowing what your ideal performance criteria are, it's really very difficult to suggest songs you might consider.

AndyG