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^[1] (the earliest known publication of the tune) calls it Hyrum. This tune is said to have been around since 1844 or before (said to have been the tune John Taylor sang with the lyrics of A Poor Wayfaring Man of Grief before the martyrdom of the prophet Joseph Smith Jr.)^[2]
Composer: Unknown (some, including the Cyberhymnal) suppose that George Coles is the composer, although a source verifying this has yet to be identified, and there is considerable reason to doubt: 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.^[3] It has since been discovered that they are not the same tune. Maybe more information was found later and not recorded (or perhaps it was recorded somewhere and we have just not yet found it).

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.