We have a couple of studies to look at for information about what exercises obtain the best contraction from the hip muscles. One study looked at muscle contraction strength of the gluteus maximus (the muscle on the back of your hip-your butt) and another study looked at the gluteus medius (the muscle on the side of your hip.
In general what we find is that weightbearing exercises (ones where you are on your feet) are better than exercises in which you are laying down. Also, anytime you can do a single leg exercise, you will get a stronger contraction out of the hip muscles on the leg you are standing on.
Above is a graph from a study looking at the muscle activity of the glute max with a variety of different exercises. As you see, single leg exercises produce the most force through the muscle. These were done with bodyweight only, so if that is easy, you can add weight by holding onto dumbbells to make to muscle work harder.
Below is a graph of muscle activity in the glute med (the muscle on the side of the hip).
Again, we see that a single leg activity (the single leg squat) and a single leg standing activity (abd/ext) produced the highest muscle activation in the glute medius muscles. Interestingly, the authors found that the sidelying clam exercise (one that is commonly prescribed by therapists and trainers), was ineffective at targeting the glute med.
So, if you want to get your hips stronger to help get over an injury or to improve athletic performance, do your exercises while standing and get on one leg. Also, make sure to add load (weight) to your exercises when you can do more than 6 reps easily if you want to really get strong!
Thanks for reading and feel free to reach out if you have any questions or need assistance.