We present a revision to the visual orbit of the young, directly-imaged exoplanet 51 Eridani b using four years of observations with the Gemini Planet Imager. The relative astrometry is consistent with an eccentric ($e=0.53_{-0.13}^{+0.09}$) orbit at an intermediate inclination ($i=136_{-11}^{+10}$,deg), although circular orbits cannot be excluded due to the complex shape of the multidimensional posterior distribution. We find a semi-major axis of $11.1_{-1.3}^{+4.2}$,au and a period of $28.1_{-4.9}^{+17.2}$,yr, assuming a mass of 1.75,M$_{odot}$ for the host star. We find consistent values with a recent analysis of VLT/SPHERE data covering a similar baseline. We investigated the potential of using absolute astrometry of the host star to obtain a dynamical mass constraint for the planet. The astrometric acceleration of 51~Eri derived from a comparison of the {it Hipparcos} and {it Gaia} catalogues was found to be inconsistent at the 2--3$sigma$ level with the predicted reflex motion induced by the orbiting planet. Potential sources of this inconsistency include a combination of random and systematic errors between the two astrometric catalogs or the signature of an additional companion within the system interior to current detection limits. We also explored the potential of using {it Gaia} astrometry alone for a dynamical mass measurement of the planet by simulating {it Gaia} measurements of the motion of the photocenter of the system over the course of the extended eight-year mission. We find that such a measurement is only possible ($>98$% probability) given the most optimistic predictions for the {it Gaia} scan astrometric uncertainties for bright stars, and a high mass for the planet ($gtrsim3.6$,M$_{rm Jup}$).
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