We report on the influence of surface confinement on the phase behavior and strain-induced alignment of thin films of blue phase liquid crystals (BPs). Confining surfaces comprised of bare glass, dimethyloctadecyl [3-(trimethoxysilyl)propyl] ammonium chloride (DMOAP)-functionalized glass, or polyvinyl alcohol (PVA)-coated glass were used with or without mechanically rubbing to influence the azimuthal anchoring of the BPs. These experiments reveal that confinement can change the phase behavior of the BP films. For example, in experiments performed with rubbed-PVA surfaces, we measured the elastic strain of the BPs to change the isotropic-BPII phase boundary, suppressing formation of BPII for film thicknesses incommensurate with the BPII lattice. In addition, we observed strain-induced alignment of the BPs to exhibit a complex dependence on both the surface chemistry and azimuthal alignment of the BPs. For example, when using bare glass surfaces causing azimuthally degenerate and planar anchoring, BPI oriented with (110) planes of the unit cell parallel to the contacting surfaces for thicknesses below 3 mm but transitioned to an orientation with (200) planes aligned parallel to the contacting surfaces for thicknesses above 4 mm. In contrast, BPI aligned with (110) planes parallel to confining surfaces for all other thicknesses and surface treatments, including bare glass with uniform azimuthal alignment. Complementary simulations based on minimization of the total free energy (Landau-de Gennes formalism) confirmed a thickness-dependent reorientation due to strain of BPI unit cells within a window of surface anchoring energies and in the absence of uniform azimuthal alignment. In contrast to BPI, BPII did not exhibit thickness-dependent orientations but did exhibit orientations that were dependent on the surface chemistry, a result that was also captured in simulations by varying the anchoring energies. Overall, the results in this paper reveal that the orientations assumed by BPs in thin films reflect a complex interplay of surface interactions and elastic energies associated with strain of the BP lattice. The results also provide new principles and methods to control the structure and properties of BP thin films, which may find use in BP-templated material synthesis, and BP-based optical and electronic devices.