Apart from some idea I have about older burial sites where they actually made a mound, Google tells me that it is a Jewish custom to place a stone on the grave as a memory of your visit (as flowers are not customary, even discouraged).
It would be possible, but not the most natural understanding. The word legge is closer to lay down or put down than to place in this sense. To put a headstone on a grave would be "Sette opp/reise en stein/gravstøtte på graven", literally to "set up" or "raise" a headstone.
It likely goes back to stone cairn burial. It's been practiced since, well, the Stone Age, or before. Cairn is a Scottish Gaelic word, but it was practiced pretty much everywhere that Celts or Vikings went. Even some Native American tribes did it.