Is it sufficient to waive a small amount of money that was not returned to my cousin, or must it be delivered to her despite the possibility of her rejecting it, especially since the money was from her personal allowance?

Your error — even if it occurred before puberty — does not negate the liability; it only negates the sin. The liability remains for the rightful claimant unless they absolve you, unless it was something minor and trivial, of no consequence, and it is known that the owner's soul is content with it; because Allah Almighty says: "There is no blame upon the blind, nor any blame upon the lame, nor any blame upon the sick, nor upon yourselves, if you eat from your houses, or the houses of your fathers, or the houses of your mothers, or the houses of your brothers, or the houses of your sisters, or the houses of your paternal uncles, or the houses of your paternal aunts, or the houses of your maternal uncles, or the houses of your maternal aunts, or [from houses] of which you possess the keys, or [from the house] of your friend. There is no blame upon you whether you eat together or separately. And when you enter houses, greet one another with a greeting from Allah, blessed and good. Thus does Allah make clear to you the verses so that you may understand." (An-Noor: 61). Ibn Qudamah said: "Customary permission takes the place of actual permission." It is preferable that you seek absolution from your cousin so that she absolves you of her right, or you return it to her. Your mother seeking absolution from your aunt is not valid, as the right belongs to her daughter.