One way our family (and many others I'm sure) have found to ease this problem, is to use flannelette hankies. They're so soft! And cheap to make too. And there's even a no-sew option, for those who cringe at the sight of sewing machines :-)
Anyway, if you frequent op shops like we do, you may have noticed flannelette sheets are often available there for a couple of dollars. Mine was a single sized, Australian made sheet in good nick...for $3. A single size sheet makes a lot of hankies. I didn't count, but the end result was a nice tall pile of them.
Now if you want to opt out of the sewing option, you'll need pinking shears (dressmaking scissors that cut a zig-zag rather than a straight line). Then all you do is measure (if you're that way inclined) and cut your hankies to your desired size. The zig-zag will prevent any fraying without the need for a hem.
My Nanna blessed me with an overlocker, so that's what I used. I just followed the stripes to keep my hems even, and then folded the flannelette into a triangle (so it would unfold into an even square...as pictured above).
If you want to sew with a regular machine there are several options. You could get all fancy (more time consuming) and fold over the hem and then sew a straight line, or you could just sew a zig-zag stitch around the edges to prevent fraying
This idea can also be used for making cloth toilet paper. Yes, I"m serious! Maybe use a different colour or print to the hankies though so they don't get confused! That might be a separate post for another time. :-)