A fun look at the meaning of "weiter" and what it can express as a verb prefix. Also: a few nice related words and usw. ;)
weiter, weit, soweit, weitermachen, weiterschlafen, weitergeben, weiter so, bis auf Weiteres,
The core idea of this root was:
weak, lack of tension, hanging down, gliding down
According to DWDS.de, the main etymological source I am using, it’s the root of sleep and the Latin labi, which was about weak, collapsing.
(English sources do not agree with this connection and list the root of sleep as sleb-, but the core idea of weak is the same)
The Latin labi is the origin of words like lapse, collapse, labor and elaborate.
Other members are sleep and lip.