That said, it doesn't look that Kasparov was under some special pressure to change his family name, he just thought that this could be a good idea.
That depends on what you mean by pressure, his trainer Alexander Nikitin has tried to take credit for the idea and convincing the whole family about the benefits of changing Garry's name, Botvinnik supported it wholeheartedly as well, and maybe also the Kasparov family in which he lived from his seventh year. It wouldn't be strange if a 12-year-old boy would find it a good idea under these circumstances and maybe even feel some pressure. More on the subject here:
http://www.chesscircle.net/forums/showthread.php?3377-Aryeh-Davidoff-Lev-Khariton-Kasparov-or-Veinstein-What-s

That said, it doesn't look that Kasparov was under some special pressure to change his family name, he just thought that this could be a good idea.