Anything from a brown paper bag
Create something from the following kind of bags:
I particularly suggest creating a puppet, but you can create whatever you want from it. Use its three-dimensional shape and/or its extra layers to your advantage.
Variations from a fortune teller
Fold a fortune teller and then fold it into something else, whatever you want. This idea for a challenge was actually first proposed by Ondrej.Cibulka four years ago: viewtopic.php?f=16&t=13235&hilit=fortune+teller
Conceptual origami
Fold something that you feel very strongly about, but don't use something concrete as its subject. Instead of that, try to express the very idea through the model. And you can't explain what it is, not even through the model's name. Try to make people get what you feel strongly about just through the model.
Fernando Gilgado's Under Mortgage and The Mass Production of Members of Society by me are great examples of what I'm talking about.
Boxes and/or lids
That's my most inclusive idea for a challenge. If you love figurative origami, then create a lid with a subject, like Anna Kastlunger's Cat Lid for example. If instead, you enjoy fractals and tessellations, then create something like Shuzo Fujimoto's Hydrangea Box. If you prefer modular origami, fold a box and lid like Chrissy's Modular Origami Box and Pinwheel Flower Lid.
Whatever type of origami you enjoy, try to turn it into a lid and/or box.