The propagation properties of elliptical Hermite-Gaussian light beam in a strongly nonlocal nonlinear medium are studied. As the characteristic width of response function of strongly nonlocal media is much bigger than the beam width, the response function is expanded twice with Taylor series and the series are both preserved up to the second-orderterm, and the approximative Lagrangian density function corresponding to nonlocal nonlinear Schrdinger equation is thus obtained. Based on this, the evolution equations, evolution laws of elliptical Hermite-Gaussian light beam′s every variable and two critical powers are also derived using variational method. When the ellipticities of light beam and response function satisfy certain conditions, these two critical powers are equal. When the initial power is equal to the critical power and light beam is incident at the beam waist, the elliptical Hermite-Gaussian spatial optical soliton is obtained. Further studies point out that the phase shift of elliptical Hermite-Gaussian spatial optical soliton is closely connected with media′s ellipticity and soliton′s order. With different values of the media′s ellipticity and soliton′s order, large positive, zero even negative phase shift may come into being.