Can soft-boiled eggs be reboiled and turned into hard-boiled eggs?
In terms of food safety, as long as the egg would have been safe to eat as it is (that is, it still is in its unbroken shell, has been cooled rapidly after cooking and then refrigerated), it is safe to recook.
In terms of quality, I am not sure that you will get a very palatable result. You are likely to end up with overcooked and rubbery whites at the very least.
You absolutely can. I boiled some eggs last night. Ran them under cold water and put them in the fridge. Went to eat them today and they were too soft, like, whites still runny too soft. Stuck them back in boiling water for a few minutes today and then ran 'em under cold water again. They were perfect! Delicious, perfect texture, perfect. You can absolutely re-boil them!
I just tried this and it seems to have worked out. I didn't really time how much longer I boiled it - I just kinda guessed. I suppose it depends on how under cooked your eggs are. The yokes were set on mine, but not the whites, so I didn't cook it that much longer. A couple of them cracked on the second cooking, so when I thought they were done I fished one out, cooled it off under the faucet, and opened it up to check.
I just boiled a dozen eggs following "the perfect boiled egg" recipe (which said bring to boil and cover for 17 minutes) but after cooling under cold water they were soft boiled. I put them back in pot reboiled the water and took them out after it began to boil and covered them (again) and cooled them with cold water (again). This time they were done but when I peeled them the shell broke off in small pieces that took twice (3x) as long to get off...but they were fine.
I did this today without a problem. Put cold/tepid water in the pan, took the half-done eggs out of their cold water bath, put the eggs back in the pan and brought the water up to a boil, left them in there about 15 minutes on low. Then put them in cold water bath again to make sure they would peel well. No problem at all. Eggs turned out fine.