Lucy's Lament is a funeral service owned by Lucy.
The service seems to specialize in the funeral of pets. This is first shown in "Along Came a Sister", when Leni presumably killed Lincoln's class pet spider, Frances, due to her extreme arachnophobia. Lucy offers to be the director for the spider's funeral.
At the funeral, everyone except Leni comes to pay their respects to the spider's untimely demise. Lincoln realizes that he was at fault for bringing Frances home with him, and should have taken Leni's fear into account, but the funeral was cut short, because it turned out that Frances was still alive and they mistook one of Cliff's hairballs for Frances' corpse.
In "April Fools Rules", while the service itself is not particularly mentioned by name, Lucy states that she will arrange Lincoln's funeral, fearing her brother may not survive the onslaught of Luan's merciless pranks, and invites Clyde to attend.
Lucy provides a few different types of burial cases for her business. One type is a shoe box that comes in a variety of types like sneakers and pumps, and there is also a classic coffin, like the kind vampires come out of. The shoe boxes are normal priced while the traditional coffins cost extra.
The coffins may also come in different sizes in case, as mentioned before, Lincoln wouldn't survive Prankmageddon as an example.