In this letter, we study the beamforming design for channel estimation of multi-functional reconfigurable intelligent surface (MF-RIS)-aided multi-user communications that supports simultaneous signal reflection, refraction, and amplification. A least square (LS) based channel estimator is proposed for MF-RIS by considering both the coupled MF-RIS beams and the introduced thermal noise. With the discrete fourier transform (DFT)-matrix, the MF-RIS beamforming design problem is simplified under the proposed LS channel estimator. The optimal MF-RIS beamforming design that achieves the Cramér-Rao lower bound (CRLB) of channel estimator is obtained with the proposed alternating optimization algorithm. Simulation results demonstrate the effectiveness of the proposed beamforming design in reducing the impact of thermal noise.