To support women's suffrage using ideas from the Declaration of Independence, the most relevant phrase would be “deriving their just Powers from the Consent of the Governed.” This idea emphasizes the importance of consent and participation in governance, implying that all individuals, regardless of gender, should have the right to vote and influence the laws and policies that affect their lives.
While "Life, Liberty and the Pursuit of Happiness" also supports the notion of equality and rights, the explicit connection to governance and consent is more direct in advocating for women's suffrage.