Achieving a large and tunable Rashba effect is of great significance for advancing both the understanding and applications of spintronics. Two-dimensional nanostructures that are expected to sustain large mechanical deformation provide a great platform to study the strain effect on the tunability of the Rashba effect. Using first-principles density functional theory simulations, we investigate the mechanical stability and the effect of biaxial tensile strain on the Rashba parameters of the two-dimensional BiTeBr monolayer sheet. The mechanical stability is computed taking into account both elastic and dynamical aspects. The simulated stress-strain curve shows that the BiTeBr monolayer sheet can withstand a biaxial loading stress up to 4.36 GPa with a critical strain of 17%. Further phonon dispersion calculations indicate that the structure becomes unstable due to the existence of imaginary frequencies for strains beyond 7% limiting the maximum stress to 2.79 GPa. Moreover, the electronic structures and Rashba parameters are calculated as a function of strain from 0% to 7%. The band gaps remain indirect and exhibit a linear dependence on the strain. The decrease in band gaps with increasing strains can be attributed to the different bonding characters of near-gap states that the conduction band minimum is of anti-bonding nature while the valence band maximum is mainly non-bonding. The Rashba parameters can be tuned by 33.6% from 1.28 to 1.71 eV Å following a linear dependence on strain. This enhancement can be understood from the perspectives of enhanced charge transfer and charge distribution inhomogeneity induced by strain. The intuitive understanding could be used to understand and motivate the application of various methods that can lead to charge redistribution to tune the electronic properties of Rashba materials.