To put forward without pressing. A suggestion leaves the decision entirely with the one who hears it.
Latin suggerere (to carry up from beneath, to bring to mind, to hint), from sub (under) + gerere (to carry); putting a thing forward without pressing was, at root, bringing it up gently rather than handing it over.
Putting a thing forward without pressing it. The decision stays entirely with whoever hears it, exactly as the old word only ever brought the idea up rather than delivering it.