We have determined the oxygen abundance of HE 0107-5240 from UV-OH lines detected in VLT/UVES spectra. Using a plane-parallel LTE model atmosphere, we derive [O/Fe]=+2.3, and a similar analysis of CD -38degrees245 yields [O/Fe]=+0.9. We estimate systematic errors due to three-dimensional effects to be on the order of 0.3-0.4 dex. That is, our derived O abundances are likely overestimates: effects from thermal inhomogeneities due to convection may require that the abundances should be reduced by 0.3-0.4 dex or even more. Radial velocity data for HE 0107-5240 based on high-resolution spectra show that over a time span of 373 days, the radial velocity was constant at 44.5 km s(-1), with a 1 sigma scatter of the measurements of 0.5 km s(-1). However, it cannot yet be ruled out that HE 0107-5240 is a very long period and/or low amplitude binary. These results provide new constraints on scenarios for the origin of the abundance pattern of HE 0107-5240. In particular, it seems unlikely that the large overabundances of CNO have been produced in a medium-mass asymptotic giant branch star that later evolved into a white dwarf. The oxygen abundance of HE 0107-5240 is significantly smaller than the prediction of Umeda & Nomoto from calculated yields of an similar to25 M-circle dot Population III star exploding as a supernova of low explosion energy (E-exp=3x10(50) ergs) with mixing and fallback. The scenario of Limongi et al., involving two Population III supernovae, predicts an oxygen abundance of [O/Fe]=+4.1 for HE 0107-5240, in strong contradiction to the observed value. In conclusion, none of the above-mentioned scenarios, in their present realizations, can satisfactorily explain the abundance pattern of HE 0107-5240.