Origins
A suggestion arrives from below. That is what the Latin says. Suggerere meant 'to carry from underneath' — sub ('from below') plus gerere ('to carry, to bear'). To suggest was to lift an idea up from beneath the surface of conversation, offering it indirectly rather than stating it outright.
The etymology captures perfectly how suggestions work. A direct statement sits on the table in plain view. A suggestion rises from underneath — half-visible, deniable, easy to withdraw. 'I suggest we leave' carries less force than 'we should leave', and the Latin sub- explains why: the idea has not fully surfaced.
Development
The root gerere is remarkably versatile. To digest is to carry apart (dis-gerere) — breaking food into components. To register is to carry back (re-gerere) — returning information to a record. To congest is to carry together (con-gerere) — cramming things into a blocked space. To exaggerate is to carry beyond (ex-aggerāre) — heaping something up past its true size.
Gesture belongs to the family too. A gesture is something carried out — an action performed with the body. In all these words, the act of carrying remains. English inherited an entire vocabulary of movement from one Latin verb, each word specifying the direction the carrying takes.