We study the sensitivity of the methods available for abundance determinations in HII regions to potential observational problems. We compare the dispersions they introduce around the oxygen and nitrogen abundance gradients when applied to five different sets of spectra of HII regions in the galaxy M81. Our sample contains 116 HII regions with galactocentric distances of 3 to 33kpc, including 48 regions observed by us with the OSIRIS long-slit spectrograph at the 10.4-m Gran Telescopio Canarias telescope. The direct method can be applied to 31 regions, where we can get estimates of the electron temperature. The different methods imply oxygen abundance gradients with slopes of -0.010 to -0.002dex/kpc, and dispersions in the range 0.06-0.25dex. The direct method produces the shallowest slope and the largest dispersion, illustrating the difficulty of obtaining good estimates of the electron temperature. Three of the strong-line methods, C, ONS, and N2, are remarkably robust, with dispersions of ~0.06dex, and slopes in the range -0.008 to -0.006dex/kpc. The robustness of each method can be directly related to its sensitivity to the line intensity ratios that are more difficult to measure properly. Since the results of the N2 method depend strongly on the N/O abundance ratio and on the ionization parameter, we recommend the use of the C and ONS methods when no temperature estimates are available or when they have poor quality, although the behaviour of these methods when confronted with regions that have different properties and different values of N/O should be explored.