* Elegy: A poem lamenting the death of a person, often with a mournful tone.
* Lament: A poem expressing sorrow, grief, or regret.
* Dirge: A mournful song or poem, often sung at funerals.
* Wailing: A poem expressing intense sorrow or grief.
* Threnody: A lament for the dead, often with a musical quality.
* Mourning song: A poem expressing grief and sadness.
* Elegaic: A term describing a poem with a melancholy tone.
It's also possible to use more general terms like sad poem, depressive poem, or poem of sorrow.
Ultimately, the most accurate name for a melancholy poem would depend on the specific themes and tone of the poem itself.