Well, there is always that association between love and death, little death and big Death.

Seriously, though, the best gloss I've found on the connection between that lamentation sort of elegy and the elegy as a love poem comes from Puttenham, who considers the elegy a sad love poem: "The third sort of sorrowing was of loues, by long lamentation in Elegie: so was their song called, and it was in a pitious maner of meetre, placing a limping Pentameter, after a lusty Exameter, which made it go dolourously more than any other meeter" (CHAP. XXIV). There you have it, "limping (and therefore sorrowful) Pentamer" and "lusty Exameter," death and love.

 

In a letter to Henry Wotton circa 1600, Donne expressed nervousness about the circulation of his writings, noting that "to my satyrs there belongs some feare and to some elegies and [paradoxes] perhaps, shame." Donne is clearly referring here to the erotic poems that were later labelled "Elegies" in the posthumous print editions of his poems. The logic of his and others' using this label is fairly simple: as Greg Kneidel points out above, the poems "read like" Ovid's Amores, which are written in elegiac meter. But it's not just that; English translations of Ovid's Amores were (as in Marlowe's translation, for example) called Elegies (on the basis of their meter). So an original English erotic poem in the Ovidian manner was also -- by extension -- an "Elegy."