Yanis' aunt, Shpetim Xhafa, wrote in a Facebook tribute: "Rest in peace my love."
The boy's 22-year-old mother and 30-year-old stepfather have been arrested and are also being questioned about allegations Yanis was physically abused.
French police are said to be working on the theory that the boy was punished by his parents for wetting his bed.
The pair reportedly called an ambulance after finding him unconscious nearly 200m from their home.
Prosecutor of Saint-Omer, Patrick Leleu, said the boy was "forced to run over a distance of several kilometers along the La Lys canal".
He added: "A child of 5 years died in suspicious conditions as to the explanations that have been given, with a medical reason not established at this time.
"The suspicious conditions of this death led us to put the parents in custody on suspicion of violence causing death."
A police spokesman told Le Parisien: "The investigation must determine whether the child died of cold, if this is the result of a fall, or a case of abuse."