What can the dummy player do in Bridge?
From Laws of Contract Bridge 2014:

Law 42 – Dummy’s Rights   Dummy is entitled to give information as to fact or Law but may not initiate the discussion, and provided he
has not forfeited his rights (see Law 43), he may also
(a) ask declarer (but not a defender), when he has failed to follow suit,
whether he has a card of the suit led.
(b) try to prevent any
irregularity **by declarer.
(c) draw attention to any irregularity, but only after play is concluded.
** Note: He may, for example, warn declarer against leading from the wrong hand.
Law 43 – Dummy’s Limitations
Dummy may not participate in the play (except to play the cards of dummy’s hand as directed by
declarer) or make any comment on the bidding, play or score of the
current deal. If he does so, Law 16 may apply. During play, dummy may
not call attention to an irregularity once it has occurred.
Dummy forfeits the rights provided in (a), (b) and (c) of Law 42 if he
exchanges hands with declarer, leaves his seat to watch declarer play
or, on his own initiative, looks at the face of a card in either
defender’s hand. If, thereafter,
(a) he is the first to draw
attention to a defender’s irregularity, declarer may not enforce any
penalty for the offense.
(b) he warns declarer not to lead from the
wrong hand, (penalty) either defender may choose the hand from which
declarer shall lead.
(c) he is the first to ask declarer if a play
from declarer’s hand constitutes a revoke, declarer must substitute a
correct card if his play was a revoke, and (penalty) unless Law 64(d)
applies, one trick is transferred to the defending side.

Contrary to the belief of many, rules are intended to keep friendly games friendly, by providing a common understanding of the game. The above are the rules for non-Tournament bridge in regards to Dummy's rights and limitations.
It sounds like Gramma is a wise old woman, who has encountered cheats in her life and wishes that her grandchildren and other relatives are never accused of such or believed to be sharp at cards. Follow her advice, and give her more credit for wisdom possibly beyond even her years.