Is it true that the angels proclaim to one whose mother has died: "Today has died she whom you used to cherish," and what are the deeds by which one may attract God's mercy after her death?

We have not come across the statement mentioned. As for the deeds by which a servant draws closer to Allah and attracts His mercy, pleasure, and love, the most important of these are performing the obligatory duties, then drawing closer through supererogatory acts, perfecting one's work, and showing kindness to people. This is in accordance with His (Allah's) saying in the Hadith Qudsi: "My servant does not draw near to Me with anything more beloved to Me than what I have enjoined upon him, and My servant continues to draw near to Me with supererogatory deeds until I love him," and His (Allah's) saying: "And do good; indeed, Allah loves the doers of good."