Is my relative obligated to compensate for the price of the golden necklace that she lost while it was in her possession, knowing that it was lost and not stolen? And is it permissible to accept compensation in the form of a gift?

Your relative is not liable for the deposit you entrusted to her, unless she was negligent or exceeded her bounds. If it was stolen from her or lost, you cannot demand its equivalent or value from her. However, there is no harm for you in accepting compensation from her as a gift, given willingly and wholeheartedly.