The welcome basket might be nice.. If you live in an apartment she might have the same problem on the other side of the wall. Plus the rooms maybe too small to switch with her child.
I would recommend to her a few things that soothe a baby. Especially a new little baby, such as the mommy heart beat bear, or the fisher price fishtank for the crib. Or maybe a white noise CD,, she could get them online for practically nothing. Or a lullabye CD that sometimes works too. If the baby is collicky she might be at her wits end and needs a friend badly. Especially if she is a single mother. Maybe the suggestions will help with the baby. Most new moms who have to go through this are already frazzled,, so I would go at it with caution,, big time.. She might be waiting to snap or cry or something. So if she gets insulted or snappy,, it's not at you,, it's at her situation. I just would hate for it to go bad for you.
Otherwise,, maybe invest in a fan, or a turn on your tv to drown out the crying. Trust me walls in apartments are paper thin,, i heard things when I lived in an apartment that I would not speak about..
Hope that helps..