Dom is Vala's son, a prostitute from Meereen working with the Sons of the Harpy. Apparently, Vala loves her son dearly and in a conversation with Varys, he uses this to his advantage to convince Vala to sail to Pentos. According to Varys, he has a breathing problem.
Dom is mentioned by Varys, who interrogates Vala about her involvement with the Sons of the Harpy in White Rat's murder. To convince her to leave Meereen, Varys brings up Dom. He tells her that he understands how Vala views Daenerys Targaryen and the Unsullied from her perspective and that he thinks that it is important that she should see things from his perspective, as well, as it is the only way how he can make her and Dom happy. When Vala claims that Varys will threaten her son instead of torturing her of information, Varys says that children are blameless and that he will never harm them. He swears to her that Dom is in no danger but, however, since Vala did conspire to kill Daenerys's soldiers, it will be possible for Vala to be executed, and Dom to be left alone. Vala claims that if she tells anything, the Sons will kill her and if she does not speak, she'll be executed either way. Varys then offers a third option where Vala and Dom can leave Meereen and sail for Pentos, instead. Varys also offers her a bag of silver to help her get started, and she accepts.
In the books
In the A Song of Ice and Fire novels, there is no mention of Dom or his mother.