Open ended engineering challenges are the best way to introduce design process thinking to students. Below you will find some ideas for challenges that provide a client to consider while designing and allow for students to create a number of diverse solutions. It is always a good idea to require labeled drawings showing their design and, depending on the activity, it can be important to emphasize that they should make a prototype that is as representative of their idea as possible even if it doesn't actually work. These types of activities are great for one time challenges to introduce your class to thoughtful design at the beginning of a UOI or as a challenge in after school.
There are MANY different versions of the engineering design process, but I find the most straight forward includes these 5 steps. The first few steps of framing the problem, brainstorming solutions, and designing one in detail are a great way to get started on any activity, even if it isn't a traditional engineering design challenge (they even work for writing a paper if you think about it). It is important to emphasize that the process is more like guidelines than a rigid set of rules and that real engineers jump back and forth between the steps as they work.
The Student Teacher Outreach Mentorship Program at Tufts sends undergraduate engineering students into local classrooms once a week to run hands on engineering activities with students. Here is a link to activities often used by the fellows in their classroom (lots of LEGO Robotics challenges), and here is a link to a bunch of useful websites for teaching engineering.
Some of my favorite shorter activities to introduce the engineering design process are: