Until the name of Amalek is erased, the name of G-d is diminished, as we see in the verse, "Since the his (Amalek's) hand is on the Throne of G-d, it is a war to Amalek for generations." The word for throne is shortened (kis instead of kisei), and the name used for G-d is Yud and Heh, which is half the name usually used (Yud, and Hey and Vov and Heh).

Rashi says in the Book of Pardes: How can G-d's Throne be missing anything? How can His Name be less than perfect? Rashi answers, that of course it is lessened. Therefore we pray "yisgadal v'yisgkadash sh'mai rabah", that G-d's Great Name be made great and be sanctified.

Maimonides discusses the importance of the sanctification of G-d's name. He describes the need for a person, the greater his scholarship, to go beyond the letter of the law and be careful not to do things that are permissible in Jewish law, such as speaking harshly, buying on credit unnecessarily, or being seen in immodest places by non-Jews.

Sanctification of G-d's name requires dying rather than bowing down to an idol. Except for murder and adultery, other mitzvoth do not require a person to lay down his life rather than transgressing.

Maimonides speaks of the sanctification of G-d's name being incumbent on all Jews. Without saying this, we would already know that Maimonides is of the opinion that Jews are required to do the commandments in the Torah. There must be a further implication in this statement. One implication is that even children, who are not required to do mitzvot under Jewish law, are commanded to sanctify G-d's name. Another implication is that when it comes to sanctifying G-d's name, it is a communal effort which can be accomplished more effectively when more and more Jews participate.

Taken from R. Shafran's Shabbos Hagadol shiur 3/27/99