The H.E.S.S. very-high-energy (VHE, E > 0.1 TeV) gamma-ray telescope system has discovered a new source, HESS J1747-248. The measured integral flux is (1.2 +/- 0.3) times 10^-12 cm-2 s-1 above 440 GeV for a power-law photon spectral index of 2.5 +/- 0.3 stat +/- 0.2 sys. The VHE gamma-ray source is located in the close vicinity of the Galactic globular cluster Terzan 5 and extends beyond the H.E.S.S. point spread function (0.07 degree). The probability of a chance coincidence with Terzan 5 and an unrelated VHE source is quite low (~ 10^-4). With the largest population of identified millisecond pulsars (msPSRs), a very high core stellar density and the brightest GeV range flux as measured by Fermi-LAT, Terzan 5 stands out among Galactic globular clusters. The properties of the VHE source are briefly discussed in the context of potential emission mechanisms, notably in relation to msPSRs. Interpretation of the available data accommodates several possible origins for this VHE gamma-ray source, although none of them offers a satisfying explanation of its peculiar morphology.