Brother is literally “neighbor,” but the reference is to one’s fellow Christian, rather than to people in general.
For his own good is literally “for that which is good,” but most translators understand this phrase in the same way as in the Good News Translation. For example, An American Translation* has “to do him good” (so also New American Bible), while the New English Bible and the Revised Standard Version have “for his good.”
It may be necessary in some languages to make please and for his own good more or less coordinate—for example, “each one of us should do for his fellow believer that which is good for him and that which will please him” or “… cause him to be happy.”
In order to build him up in the faith translates a noun phrase (literally “for building up”). Most commentators understand this as a reference to the man who is weak in faith (Moffatt “by building up his faith”; New American Bible “by building up his spirit”; the Jerusalem Bible translates both clauses by “help them to become stronger Christians”). However, the New English Bible applies this in general to the Christian community (“and will build up the common life”). In light of the context, however, the former interpretation is to be preferred.
This final purpose clause may require some special introduction based on a repetition of, or a reference to, what has just been said—for example, “we should do this in order to cause his faith to become strong” or “we should do this in order that he will become stronger in his believing.”
Quoted with permission from Newman, Barclay M. and Nida, Eugene A. A Handbook on Paul’s Letter to the Romans. (UBS Handbook Series). New York: UBS, 1973. For this and other handbooks for translators see here .
