Exactly why this name was chosen and what it might imply are matters of some discussion among fans.
The name is Romanized as "Shin-Ra" in official Japanese materials. This spelling convention, as opposed to the more straightforward "Shinra", may be simply for stylistic reasons. In the English version of Final Fantasy VII, the name of the corporation appears in dialogue as "Shinra".
In Final Fantasy X-2, a member of the Gullwings named "Shinra" talks about extracting pure energy from the core of the planet. In an interview for the Final Fantasy X-2 Ultimania Guide, scenario writer Kazushige Nojima has stated that the character Shinra is a deliberate nod to the Shin-Ra corporation of Final Fantasy VII, and implying that he envisioned the events of Final Fantasy X-2 as a prequel to Final Fantasy VII[1]. This connection was again referred to in the Final Fantasy VII Ultimania Omega Guide, published by Square Enix.