Most of my favorite hip strengthening exercises involve standing on one leg or shifting weight to one leg. Standing on both legs it's easy to hide from hip problems or weakness.
Working on one leg it's easier to detect problems and fix them.
Because I've had hip problems, a lot of my hip strengthening exercises include deliberately activating particular hip muscles. If you have hip problems, or any muscle problems for that matter, being able to turn muscles on and off at will can be used as a fault finding method. But outside of that it can also be a way of making difficult poses easier. You can play with different muscle activations to see which one helps the most. And on any given day, the activation that helps may be different.
I'd say my favorite hip strengthening exercises is balancing on one leg in a standing forward bend with the lifted leg flexed at the hip (pulled forwards).
You can start with both feet and both hands on the floor (or hands on yoga blocks or a chair if you can't reach the floor.) Knees are straight. Shift weight to one leg and lift the other leg. Pull your lifted leg forwards.
If you have difficulty pulling it forwards your first exercise can include reaching the leg back and up as high as possible and then pulling it forwards. Do both movements slowly and smoothly. Repeat a few times then switch legs.
From here the next step is to keep the lifted leg pulling forwards and lift the hands. To do that shift your weight back so that you press evenly through heel and forefoot. Then lift your hands. Pause, then replace your hands on the floor and shift your weight forwards. Repeat a few times before changing sides.
Prior to shifting your weight back and lifting your hands you can activate the hip of the standing leg. Or activate both the hip and the knee. You can relax both after you shift your weight forwards.
You could also activate the hip of your lifted leg to help pull your lifted leg further forwards.
Another knee and hip strengthening exercise is half moon pose.
For this, the exercise can be to activate the knee and hip, then relax. You could make the hip feel strong or focus on the glute.
The next step is to add a weight shift.
Make your knee and hip strong, then shift your weight to your foot, then lift your hand. Pause, put the hand down and shift weight midway between hand and foot and then relax.
Usually I'll rest from one foot exercises by doing regular standing exercises. The focus could be on using friction or simple muscle activation to strengthen the legs.
Another hip set of hip strengthening exercises that I sometimes use to finish a class with involves controlling the side tilt of the pelvis. Standing on one leg with the other leg lifted, the idea is to use the hip muscles of the standing leg to control and vary the tilt of the pelvis. This mainly exercises the glutes but it can also be a means of devoloping better pelvic awareness.
Standing upright on one leg with the other foot slightly lifted focus on lifting the hip on the lifted leg side, then sinking it down. Focus on using your standing leg hip muscles.
Switch legs and repeat.
Next try the same exercise but bent forwards with the torso horizontal. You can do this hip strengthening exercise with the hands on the floor (or on a chair or on yoga blocks) or with the hands lifted.
You can repeat the exercise in tree pose focusing on the lifted leg hip.
Then finally try it in a variation of eagle pose. Here, cross the knees but don't bind the feet. Instead as you lift the hip move the lifted leg shin towards the standing leg.
Conscious Muscle Control videos: Friction and Pressure includes more leg and hip strengthening exercises. Some of the exercises use friction and others pressure to make muscle activation (and muscle awareness) easier to learn.
Why improve muscle control?
Muscle control not only helps you to control your body, it also helps you to feel it.
Muscle activation creates the tension that not only moves your body, but helps you to "sense" it.
With better muscle control you can use your body with less effort and make it easier to balance, improve flexibility and deal with pain and poor posture.