Is mentioning the reason for distributing prayer beads—that it is a charity for the souls of the grandfather and grandmother—considered showing off, and is the relatives' knowledge of this also considered showing off?

Giving charity on behalf of the deceased is permissible. The intention of the heart suffices for this, meaning that its reward is for so-and-so, without needing to mention their name or utter it aloud, for Allah knows intentions and purposes. It is permissible to use prayer beads (misbaha) and a manual counter for glorification. As for your relatives knowing about your charity for your grandfather, whether it constitutes showing off (riya') or not depends on your intention. If your heart turns to their praise, then it is showing off, but if it does not, then it is not.