Brich Hu or Amen?
There are a number of different customs as to how one should respond to the "brich hu" in the kaddish. According to some customs, the proper response is "brich hu", while others respond "amen." Others have the custom to respond "brich hu l'eila min kol…" and actually recite the entire remaining section as part of their "brich hu" response. We will explore the development of these customs.
According to the Shulchan Aruch, the proper response when hearing "brich hu" is "amen", and this is the custom of the Sefardim, Chabad, and a sprinkling of others based on the teachings of the Arizal. The Rema, however, writes that responding "amen" is not appropriate at this point in the kaddish, as he subscribes to the view which teaches that there should be no interruption whatsoever between the words "brich hu" and "l'eila". According to this view, "brich hu" is actually intended to be a part of the words which follow it, not those which preceded it. It may just be that the Rema opposes any response or interruption whatsoever at this point in the kaddish, whether from the leader or the congregation. Indeed, there are a small number of congregations in which no response at all is offered at "brich hu", a custom apparently attributed to the Gra. There is also a view which says that "brich hu" actually belongs to the words which precede it, not those that follow it.
The Kaf Hachaim writes that the Arizal's view is that "amen" is the proper response to "brich hu", and that this and the other "amen's" of kaddish are all vital and strategically positioned responses which serve to properly separate between the different components of the kaddish. The Shaarei Teshuva explains and supports this view, as well. Rabbi Joseph B. Soloveitchik also subscribed to this view and would answer "amen" at "brich hu" in kaddish.
It is interesting to note that the source for responding "brich hu" in kaddish is actually quite obscure and is not clearly acknowledged anywhere. It seems to derive from the Likutei Maharich, based on the Taz, who writes that answering "brich hu l'eila" is intended to serve as a prompt to the one reciting kaddish that his "brich hu" is connected to the following segment which continues with the words "l'eila min kol birchata". Such a response, however, posses a problem as one can be misled to believe that God is only to be praised "l'eila", above in Heaven, and not on Earth below. Therefore, the word "l'eila" was dropped and the response was left simply as "brich hu". It seems that it was only a congregational response of "brich hu l'eila" which was deemed problematic, however, the custom of some ba'alei tefilla to recite the words "brich hu l'eila min kol birchata" in a single uninterrupted utterance when reciting Kaddish does not pose such a problem.
Rabbi Mordechai Winkler writes that the congregational response of "brich hu" is the true Ashkenazi custom, with "amen" truly belonging to the Sefardim. It is explained that according to the view of the Rema, which is that the leader should recite the kaddish as "brich hu l'eila min kol birchata" in a single uninterrupted utterance, responding "amen" would serve no constructive purpose at all since no "interruption" should be made at that point! Responding "brich hu", however, is more consistent and it can be viewed as a concurrent prayer rather than an interrupting response. The Eliyahu Rabba subscribes to this view. It should be noted that "brich hu" is actually one of God's names.
Responding "amen" is especially fitting for those who follow the customs of the Arizal in which an interruption between "brich hu" and "leila min kol birchata" is required. It seems, however, that the Mishna Berura would prefer the response of "brich hu" over "amen" even by those who otherwise follow the Arizal.
It is interesting to note that it is not so clear that it is permissible to respond "brich hu" when hearing kaddish while in the midst of pesukei d'zimra, as it is deemed by some authorities to be an unwarranted interruption. It goes without saying that this ruling is the same for one who hears kaddish while in the midst of "birchot kriat shema", and even when "bein haprakim". Indeed, Rabbi Mordechai Willig is of the opinion that one does not answer "brich hu" to the kaddish before barchu even when has finished reciting pesukei d'zimra. One should always be careful when reciting or responding to kaddish to say "brich hu" properly and not accidentally say "brichu".
 Aruch Hashulchan O.C. 56:6
 Be'er Heitev 56:7
 O.C. 56:2
 Or Zarua 1:42, cited in Piskei Teshuvot 56:7
 Rema O.C. 56:2
 Ma'aseh Rav 54, Biur Hagra O.C. 56
 O.C. 56:29
 O.C. 56:6
 Aruch Hashulchan O.C. 56:6
 This paragraph based on Minhag Yisrael Torah O.C. 56:1
 Levushei Mordechai 2:10, cited in Minhag Yisrael Torah O.C. 56:1
 Berachot 19a
 Kaf Hachaim O.C. 56:29
 Shaar Hatziun 56:30
 Igrot Moshe 2:16, Tzitz Eliezer 11:3
 Kaf Hachaim 56:14