Okay first of all, it is not the "Nurse" who should be telling you this.
She is not a Pediatrician.
2) Your family, genetically, has short people. Stature/size of a person/child, is largely due to, genetics.
3) You said, that so far your son's size has not "bothered him terribly." So why, worry.
4) All children have growth spurts. BUT it is not the same for ALL kids and it is NOT the same for each gender. Kids, grow differently and have DIFFERENT physiques.
The "size" of a boy... VARIES. Boys, do NOT have to be all big and thick and bulky and huge, in order to be, a boy.
And growth spurts do not occur at the SAME age for ALL and each and every kid.
5) You said, your son does not have medical issues. He is, short/small/lanky or whatever it may be. Fine.
6) Do not give your son a hang up about his size. MANY of my kids' classmates, both boys and girls, are tiny. They do NOT have any social problems nor self-confidence problems because of it. My daughter who is going to 6th grade, has a classmate that is tiny. In fact, my 6 year old son, is taller than he is. So what. That classmate is fine, has many friends, does not get teased, does not have a hang-up about his "size" and is just a normal boy and he is involved in many sports and does just fine as compared to other boys who are bigger, than he is.
7) Then, I have a cousin, who is SUPER tall. His parents are not. They are small. But he for some reason, REALLY shot up in height, like in High School. Everyone jokes, that maybe way back in his family's gene pool, there was someone who was tall. Or, it is just the way he is. Fine. Whatever.
8) IF you even, consider "growth hormones".... it is NOT the Pediatrician that decides. The Pediatrician, SHOULD be referring you to a Pediatric Endocrinologist. But again, your son does not have health/medical issues. You just think he is small. To me, that is not a reason... to put a kid on growth hormones. And a Specialist, may not even give it to him, just because there is no medical reason, to do so.
And again, it is NOT the "Nurse" who should be telling you this or what he should take, for his size.
9) And, growth hormones, has side effects. Like anything. And is not a magic wand.