The best elevation data as of 2016 seems to be WorldDEM, although I haven't seen any application in economics research. It's also not for free of charge. Below is the list of other elevation datasets (available for free of charge) that have been used by economists in the past.
Developed by the U.S. Geological Survey's Center for Earth Resources Observation and Science (EROS) in 1996, GTOPO30 provides elevations at the 30 arc seconds (roughly 1km) grid level. See the USGS/EROS website for detail.
GTOPO30 was used by Deininger and Minten (2002), Nunn and Puga (2007) to measure the degree of ruggedness of the earth surface of each country, and Duflo and Pande (2007) to calculate river gradient in India.
GTOPO30 is now superseded by Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010).
SRTM3 / SRTM30
SRTM3 is an updated version of GTOPO30 (I suppose) at a higher spatial resolution of 3 arc-seconds (roughly 100m). SRTM30 is a version that aggregates SRTM3 to the 30 arc second resolution. SRTM30 is supposed to be better than GTOPO30 (see the description of SRTM30 on this page).
For SRTM3 (version 2.1), the data is available here and the documentation is available here. For SRTM30 (version 2.1), both the data and the documentation is available here. For a graphical interface to download the data, visit here.
SRTM30 has been widely used by economists: Taryn Dinkelman's working paper (now published in American Economic Review) "The Effects of Rural Electrification on Employment: New Evidence from South Africa" (to create an instrument for electricity grid placements); Melissa Dell's working paper (now published in Econometrica) "The Persistent Effects of Peru's Mining Mita" (to create control variables); Acemoglu and Dell's paper forthcoming in AEJ Macro "Productivity Differences Within and Between Countries" (to calculate the distance to paved roads that takes into account elevation); Olken (2009) (to obtain the strength of TV signals in each sub-district of Indonesia); and Yanagizawa (2009) "Propaganda and Conflict: Theory and Evidence from the Rwandan Genocide".
How to use GTOPO30 / SRTM30 in ArcGIS
Here is the tip for "How to import GTOPO30 or SRTM30 data into ArcMap (for ArcGIS 9.x)". Step 7 should be skipped because the SRTM30 version 2 uses the value 0 (instead of -9999) for the ocean (see section 1.2 of the documentation).
ASTER Global Digital Elevation Model Version 2
An alternative elevation data to SRTM. Rexer and Hirt (2014) validate SRTM and ASTER against elevation data in Australia, concluding that SRTM is superior in general, with ASTER better for mountainous areas.
Used by Mariaflavia Hariri's working paper entitled "Cities in Bad Shape: Urban Geometry in India".
Elevation data is also available by TerrainBase, constructed by National Oceanic and Atmospheric Administration (NOAA) and U.S. National Geophysical Data Center (downloadable at the Atlas of Biosphere). This one is used by Michalopoulos (2008). It is not clear if this is the same as, better or worse than, GTOPO30 and SRTM30. However, if the study area is the whole globe, this data is easier to use because it comes in one file. (GTOPO30 and SRTM30 are provided in several files each of which covers a part of the whole globe.)