The CoCrFeNi high entropy alloy (HEA) with face-centered cubic (FCC) crystal structure exhibits excellent ductility values even at cryogenic temperatures. However, since this HEA is relatively weak in strength, it may not meet the requirements of industrial applications in terms of strength-ductility trade-off. Therefore, the systematic addition of yttrium (Y) into CoCrFeNi HEA was investigated in the present study to increase the strength by solid solution and second phase strengthening. The HEAs were produced by vacuum arc melting, suction casting, and subsequent homogenization at 1150 °C for 24 h. The structural development of the HEAs was investigated by using the X-ray diffraction (XRD) technique revealing the formation of a solid solution phase and Ni3Y-type hexagonal structure (HS) second phase. The corresponding microstructure of the HEAs was examined under a scanning electron microscope (SEM) revealing the transformation of the microstructure from elongated grains to nearly equiaxed grains with the increase of Y content from 2 at. % to 4 at. %. The mechanical properties of the HEAs were investigated by using hardness and compression tests. The results exhibited a dramatic increase in the hardness from 143 (±2) HV to 335 (±7) HV and in the yield strength from 130 MPa to 1025 MPa with 4 at. % Y addition. Our study has revealed that the addition of rare earth Y element results in further development in the strength of the CoCrFeNi for potential engineering applications.