General Information

The actual name of this tune has yet to be verified, however the Cyberhymnal calls it Man of Grief (without a source, and so perhaps they coined it), and the Latter-day Saints Psalmody, 1889, no. 34 (the earliest publication of the tune) calls it Hyrum.
Composer: Unknown (some, including the Cyberhymnal) suppose this to be George Coles, although a source verifying this has yet to be identified. The confusion appears to have been started by an assumption that since the tune Duane Street was not found to compare it with the tune at hand, and Duane Street was known to have been associated with A Poor Wayfaring Man of Grief, that perhaps they were the same tune. Maybe more information was found later, but if so, the public has yet to know it.^[1]

Latter-day Saint Information

This is the tune used in the current (1985) LDS hymnal with A Poor Wayfaring Man of Grief, although that hymnal refers to the tune as Duane Street (which is another song entirely, though also associated with A Poor Wayfaring Man of Grief). Here is a source that gives hints as to how this confusion may have begun, as mentioned above also.

↑ Stories of Latter Day Saint Hymns Their Authors and Composers, 1939 (p. 12)
↑ Hymn no. 34 (A Poor Wayfaring Man of Grief) in the Latter-day Saints Psalmody, 1889 says these lyrics (no. 321) may also be used with this tune.