Good question- according to the American College of Sports Medicine, a good warm up will get your heart rate to about 50-60% of maximum. This increase in HR will increase blood flow to skeletal muscle and joints (and to address your question in the comments, if you are doing almost any kind of warm up, that should put your joints through plenty of ROM... getting more ROM is not a good strategy for you). For someone who has hypermobile joints, I would do a longer warm up (try 20 minutes), and definitely avoid stretching (http://www.theptdc.com/2013/10/why-you-must-not-stretch-hypermobile-clients/). With hypermobile joints, make sure not to lock your knees or elbows during movements such as stepping or jogging, and do not exaggerate ROM during exercise. Closed chain exercises (where feet are on the ground) are recommended. I would also avoid plyometric exercises like jumping, leaping, etc.
- A good warm up for you might be closed chain exercises that keep you in your mid-range of motion, like step-ups, light bodyweight squats, wallsits, throwing in a little arm movement such as clapping or flapping motions, keeping good control of your ROM, staying low, and avoiding aggressive movements.
You are definitely right to be wary of stretching- the myth of stretching before and after workouts is unfortunately widespread throughout the fitness industry. In many of my classes in preventing athletic injuries, the consensus was that stretching is only necessary to achieve a ROM which allows you to perform necessary activities to YOUR life. Meaning, if you're a cyclist and you don't need to get your arms behind your back, there is no need to stretch that way. You can actually do more harm than good by overstretching and stretching more than necessary.
Bottom line: if it doesn't feel right, don't do it!
Hope that helps!