From SAGE to Sage to SageMath
SAGE -- Software for Arithmetic Geometry Experimentation
In 2004, William Stein started work on SAGE, which stood for "Software for Arithmetic Geometry Experimentation".
In 2005, version 1.0 was released.
From SAGE to Sage
As the scope of SAGE grew to cover more and more of mathematics, the acronym was dropped and the name became Sage, with a recommendation to not use the all-capitals spelling.
Please write Sage rather than SAGE, unless specifically mentioning the early history of the system.
The internet domain reserved for Sage was sagemath.org.
SageMath -- the Sage Mathematical Software System
In 2015, it was decided that "Sage is grownup, needs a full name", and the name was changed to SageMath. The prompt remains "sage:" and the short name keeps being used when there is no ambiguity.
For other things called Sage, see ThingsCalledSage.