I see previous responses have mentioned ADD, the non-hyper variety, testing, and medication. While your son might not be "diagnosable" as having ADD, I believe, based on your description, that he has AD/HD-primarily inattentive type.
Takes one to know one. I too had trouble paying attention in school. I was never diagnosed because I managed to get straight A's in school, so no one ever thought I had a problem. But I did. I slaved over my homework to try to learn what I'd missed in class AND complete the homework. I spent many more hours doing homework than most of my classmates.
In class, I would daydream, think about social issues with which I was struggling, notice all the fidgets and naughties of other pupils, and generally not pay attention to the lessons. My attention would snap back when the teacher asked someone a question, because that stressed me. I was always grateful it wasn't me who had been asked the question. I would not have known the answer, and would have been embarassed, humiliated, and felt stupid, *again*. After a question snapped my attention back to the teacher, for a few minutes I was under what I call "optimum stress" and could focus on the lessons before drifting off again as the stress faded. I did best with teachers who asked classmates questions frequently.
In college, I often nodded off to sleep in class, because the lecturers did just that - lecture - and rarely ever asked questions of students. I self-medicated with No-Doze (high-dose caffeine pills) in order to survive in college. Many years later, I learned about AD/HD and self-diagnosed.
The use of stimulant medications helps arouse the AD/HD brain into the "optimum stress" zone where the prefrontal cortex can function. So do various risk-taking behaviors, and so does procrastination until one is facing a deadline. The AD/HD brain needs more stress than healthy brains in order to get started, to regulate impulses and to pay attention. Yet it also becomes overwhelmed at lower levels of stress than a healthy brain. So, a person with AD/HD has a narrower window of environmental stress in which s/he can function well.
Hormonal fluctuations affect the AD/HD brain more than they do a healthy brain, and AD/HD children are more often involved in riskier behaviors as teens than their peers. Your 13-year-old son is at a good age to begin treatment. If you start now, you will likely help him steer clear of the self-undoing, self-medicating and esteem-damaging teen mistakes so common among young people with AD/HD.
Besides medication and self-inflicted stressors, biofeedback can help your son permanently improve his functioning in public and social situations. The costs seem prohibitive, but when you view the costs of lifelong medication (including side effects) and ongoing psychiatric supervision, biofeedback looks like a bargain.
If your son is never diagnosed, that's actually a gift to him. He'll never label himself, and the all-powerful insurance companies won't label him either. One need not be "diagnosable" to get biofeedback. So, check into biofeedback sessions for your son. It's essentially like playing video games, but of course there's a ton of science behind the scenes. Your son will probably enjoy the sessions; they're painless, and over the long haul he will be much, much better off than if you either medicate him or tough it out without treatment.