From SAGE to Sage to SageMath
This page provides some more detail on the evolution of the name.
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 0.1 was released.
As the scope of SAGE expanded, it became an acronym for "Software for Algebra and Geometry Experimentation".
From SAGE to Sage
As the scope of SAGE grew to cover more and more of mathematics, the acronym was dropped altogether 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 last name", and the name was expanded to the "full name" SageMath, with the short name "Sage" kept in use when there is no ambiguity, and the prompt remaining "sage:" in the Sage REPL.
For other things called Sage, see ThingsCalledSage.