The giant magnetostriction of the Bi2Sr2 – xLaxCuO6 + δ (La-Bi2201) single crystal at x = 0.8 is first detected for the low-temperature region. This composition is at the insulator/superconductor interface of the phase diagram of the La-Bi2201 high-temperature superconductor (HTS). The sample is heavily underdoped, nonsuperconducting La-Bi2201, and previously exhibited anomalous magnetically dependent negative thermal expansion at low temperatures. We observe the giant magnetostriction of ~10–4 known for superconducting HTS cuprates. The effect is observed in the temperature range from 6 to 70 K. The results point to the residual fluctuating superconductivity which is observed in HTS to high temperatures.