We should note that it wasn't JUST a name change. But part of the total reorganization of the defense establishment.

In particular it wasn't just War getting a name change, but the consolidation of previously co-equal departments with the Department of the Navy. Previously the only person an Admiral and General both answered to was the President, and everything below that was a weird balance and compromise without anyone else to step in to force a choice or defuse a conflict. While you also had the consolidation of the various one off and newly created support agencies, and other areas that could be overseen directly by the new level of oversight, spawning agencies like DARPA, DIA, DLA, etc today.

We should also note that the first name if the new group was the "National Military Establishment" as it was meant to be the new joint organization between the co-equal Army(renamed from war as they were not the only dept involved in fighting any war, and of course being seen as the savior of freedom and not an aggressor looking for a fight has value even then), Navy, and Air Force. But it was later made more clear, and of course brought in line with the rest of the govt nomenclature by being changed to Dept of Defense a few years later.