Happy baby hip stretch is a reclining (laying on your back) hip stretch that I like to use as a prep for foot behind the head yoga pose.
I often do it after pigeon yoga hip stretch.
Where in normal yoga Happy Baby pose your grab both feet, in this variation you only grab one foot.
Start with both knees bent. Grab a foot with both hands, bend the knee and use your arms to pull the knee to the floor to the side of your ribcage.
Keep your shin vertical.
You can focus on relaxing your hip.
Have your other knee bent close to the chest to start with and then slowly straighten that knee and reach the foot forwards. This adds weight to same side hip helping to push it down towards the floor.
To better prepare for foot behind the head, hold the foot with the same-side-arm and reach the other arm to the side and place it on the floor.
You could try adding more wieght by moving your non-stretching leg to the side (though this can be difficult.) Keep that foot lifted, or if you do have it on the floor, then pull up on the leg so that the same side hip continues to sink down.
This pose is very similiar to low lunge yoga pose.
It's also similiar in effect to the marichyasana style yoga poses.
In all of these poses you are trying to get the knee past the plane of the upper back. The difference is that in these poses the torso moves relative to the leg.
In Happy Baby Hip Stretch you move the leg relative to the torso.
I mentioned above that you can focus on relaxing the hip. An option is instead to make it active. In this case, push the foot against the hand even as you use your hand to pull down on the foot.
You'll be using your hip muscles against your arm muscles. And that will give you some control over your hip (as well as your arm.). You may find that you can strongly resist using your hip. Then, while keeping the resistance, you may also find that you can allow your knee to slowly move towards the floor.
It's a lot like squatting down slowly. When you squat slowly, your quads and glutes are slowly lengthening even as they resist the weight of your body sinking down. In happy baby, you can resist using the hip muscles while allowing them to slowly lengthen.