Word for song sung at a burial?

A dirge is such a song.

a lament for the dead, especially one forming part of a funeral rite.

a mournful song, piece of music, or poem.

-Oxford Living Dictionary

Example:

A melancholic dirge was sung by the mourners at the burial of the youngest son of the mayor.

"elegy" is also appropriate.

2. a : a song or poem expressing sorrow or lamentation especially for one who is dead

Source: https://www.merriam-webster.com/dictionary/elegy

I think you could use "requiem".

According to the Merriam-Webster online Dictionary, the second meaning of "requiem" seems to be what you are looking for:

Definition of requiem 

1 :  a mass for the dead 
2 
  a :  a solemn chant (such as a dirge) for the repose of the dead 
  b :  something that resembles such a solemn chant 
3 
  a :  a musical setting of the mass for the dead 
  b :  a musical composition in honor of the dead

Source: https://www.merriam-webster.com/dictionary/requiem

A lament is a suitable word also - a lament implies sadness, grief, sorrow - a dirge also implies miserable and slow.