Optical and ultraviolet spectroscopy obtained with the Hubble Space Telescope Faint Object Spectrograph at three locations in the central H II complex of the amorphous starburst galaxy NGC 5253 confirms an apparent area of enhanced nitrogen abundance seen in ground-based studies. At two positions, N appears enhanced by a factor of 3 [log (N/O)= -0.85] compared to a third location 50 pc away where the measured N abundance is typical of metal-poor galaxies [log (N/O) = -1.30]. No other elemental species shows spatial abundance fluctuations, including C as measured from the C III] lambda lambda 1907, 1909 Angstrom emission lines. Extinction to the emission-line regions is quite low (A(V) = 0.25), and we rule out reddening uncertainties as the cause of the apparent N enhancement. Comparison of the observed ionic fractions to predictions of photoionization models shows that ionization uncertainties cannot be responsible for the anomalously high N/O ratio. These findings require the existence of a N production mechanism that is decoupled from C and O production. Although the high N/He enrichment ratio is consistent with that expected from non-Type I planetary nebulae (PNs), the localized nature of the enrichment, the large number of PNs (150-500), and the long timescales (10(9) yr) required make this explanation highly unlikely. Given the proximity of the N overabundance to a very young, heavily obscured star cluster, we attribute the N excess to recent ''pollution'' from massive stars. Plausible N enrichment scenarios involve O star winds, He-deficient W-R star winds, and/or ejection events from luminous blue variables. If the N enrichment is due to localized pollution from the winds of W-R stars observed in the burst, then an accompanying He enrichment of 40% ought to be observed, but we are able to rule out He enhancements at the 2 sigma level at both locations showing N enhancement. We propose that if the N-rich wind or ejecta is incorporated into self-gravitating clumps of molecular gas, and if the 10(6) yr old clusters of massive stars in starburst galaxies are precursors of globular clusters, then N-overabundant halo and globular cluster stars in the Galaxy may owe their chemical peculiarity to similar N-enrichment episodes early in the history of the Milky Way.