Ishmael Osamor, 29, pleaded guilty to drug offences in September and was given a community sentence.
He was found with £2,500 worth of drugs at a music festival. Ms Osamor has continued to employ her son in her parliamentary office following the conviction.
She claimed that she knew nothing about the case until after the sentence was handed down on 19 October. However, it was revealed that Ms Osamor wrote to the judge in the case asking for leniency.
The Times reported Ms Osamor told a journalist who went to the MP’s home that she “should have come down here with a bat and smashed your (the journalist’s) face in”.
In a statement posted on her Twitter feed, Ms Osamor said was stepping down to "concentrate on supporting my family at this difficult time".
Labour leader Jeremy Corbyn said: "I have accepted Kate Osamor's resignation and would like to thank her for her work as our shadow secretary of state for international development.
"She brought a new dimension to the role by committing Labour to tackling global inequality as well as poverty as part of building a world for the many not the few.
"I know Kate will take this time to support her family, work for her constituents and support our party's efforts to rebuild Britain from the backbenches."