How it all started
Sometime around 1997-98, when the draft of the first C++ Standard was being finalized for publication as an ISO/IEC Standard, Robert Klarer from the IBM Labs conceived the idea of a programming language that would be called BOOSE (pronounced "booz"), and which would compete with Java in the area of high-performance embedded software development, which the latter had been aimed at. In a 1998 article for the now defunct C++ Report magazine, C++ guru Herb Sutter wrote a tongue-in-cheek spoof on this new language, whose name ostensibly expanded to Bjarne's Object Oriented Software Environment. In this article, he claimed that portability and potability were, among other things, key advantages of this language, which also supposedly promoted extraordinary camaraderie in team environments and made developers excessively happy, communicative, and passionate.
While this was an April Fools' Day article in 1998, the fact remained that the first C++ Standard was going to have a fairly basic standard library consisting of a memory allocation subsystem, type-generic containers and algorithms, a string class, basic abstractions for input and output devices, and sundry utilities. Now around the same time, a few folks from the C++ Standards Committee formed a group that worked on producing a collection of high-quality, peer-reviewed, free, and open source libraries in C++ that would have wide applicability and complement the features in standard C++. Inspired by BOOSE, perhaps for its stated competition with Java, which was a newer language but with a much richer library, they named this initiative Boost, a working title that stuck (source: FAQ on the Boost website, http://www.boost.org).