Is it permissible to give charity on behalf of a sick son to one of his relatives, such as a brother, and does it have to be monthly or can it be annual?

There is no objection to giving charity on behalf of the sick, as there are narrations indicating its desirability, even if the hadith is weak. Charity is generally encouraged, and particularly so during times of affliction, especially if given to a poor relative. The questioner may give charity to his relatives with the intention of giving it on behalf of his sick son. It is not required for this to be monthly, annually, or time-bound; it is permissible to give charity once or to repeat it.