Another vote for a refurbished Lenovo, get one with SSD. But there are some downsides to consider...
Up until this year all of our home computers have been refurbished Lenovos and most of our more junior guys at work also get them--so I have LOTS of experience with them. I have been using various x series at work for a good 16 years...
The biggest problem is the battery. A good chunk of the battery's lifespan has been used up by the previous user (lease). This may matter for a student if they will be using them in class and need many hours of battery life. IMO you get about 50% to 80% of new (depends how it was used before) when you get it but expect either to mostly be plugged in or replacing the battery in two to three years, the genuine batteries are $$$$ and the aftermarket ones don't tend to last long before they are short on power. The older laptops are also more power hungry than many of the newer ones. My X1 is now just over a year old (not a refurb) and I still get almost a work day out of the battery but it is down one or two hours from new.
If you go back to I think the #40 models (eg x240, T440) they did away with the mouse buttons and made them part of the pad. If you use the nub they were absolutely awful. The next models they went back to separate buttons due to complaints. Avoid these if you use the nub, otherwise they will need to carry a separate mouse (what I did with my last laptop).
For portability I really like the x series as they are light and fit nicely into back packs. I have an obusform laptop backpack and one of the costco roots specials, both work very well. Roots is much bigger and they last a long time. They usually have them in August, back to school...
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This year I went new for the kids and wife. Lower end HP ($400 on sale) for my oldest (grade 9), not super prociessing powerful but it does more than she needs, she is happy. Other kid is in grade 7 so chromebook, again going well. My wife really wanted a "new" computer so we dropped a grand on an Acer (on sale), she is using her old Lenovo still about 70% of the time, refurb we have had for at least 4 years, it has been dropped a few times, screen has dead spots now and it needs to be plugged in....