Over the many years of my teaching, I have read sources on how to begin a lesson. I remember reading one that discouraged the teacher from commenting on anything that is not musical, specifically a new article of clothing that the student is wearing. This person's approach was to get right to the task at hand, waste not, want not.
Right now I am reading With Your Own Two Hands by Seymour Bernstein. His approach is quite different as he acknowledges from the onset that teaching is also psychology. The same thing was said by my long-time teacher and role model, Galen Deibler. By psychology these teachers really mean empathy and relationship building, not analyzing or labeling students. What they mean is connecting, and taking some time to connect with a student may be the best use of the first few minutes of a lesson.
Connecting can happen on many different levels. Checking in with a student as to how the day is going is one simple way. A student who is coming right from school or another activity may need a few minutes to decompress before being able to shift gears for a music lesson. A student who is feeling a bit under the weather, especially the headache sufferers, will benefit from your knowing and adjusting the lesson accordingly.
Finding out about your students interests outside the lesson is a good way to get to know the whole student. First of all, you will know how they spend their time, but also you may be able to reference other activities in relationship to what is happening at the piano. Some sports and activities like dance may have movements that the student can learn to carry over into piano technique. I have also learned some very interesting bits of information from students with interests different from mine, and they have a few minutes to be in the teacher's seat during the lesson.
Some teachers use a process of matching to connect with students. This process involves matching the student's energy level at the beginning of lesson, then bringing the level to one that works best for the lesson. Again, if you know your student, you may find this helpful. A student who has challenges with self-soothing may do better to be met with a calm and quiet demeanor rather than upping the emotional ante with equal levels of excitement.
Quiet listening can be the most useful way to connect with some students. These students don't want an easy answer or even the pat response, "I know how you feel." Maybe you do, maybe you don't. Consider listening for a minute or two as a way to establish both safety and connection.
Setting common goals is another way to connect. This aspect overlaps with the need for control. Students who want to control all aspects of the content of the lesson probably need to dial back a bit. And students who present inappropriate goals will be frustrated, turning connection into disappointment.
However you connect with a student, even if you prefer to avoid complimenting new clothes or hairstyles, remember that connection is key to a successful student/teacher relationship.