Is the questioner obligated to lend her sister money to pay her son's dowry, knowing that her sister and her family are well-off, and that the questioner is hesitant to do so?

If you have money that you do not need, and your sister is able to repay it, and you do not fear her procrastination, then it is advisable to lend it to her; for it involves upholding ties of kinship and seeking Allah's reward. A good loan is a righteous deed for which Allah grants the greatest reward, and it entails upholding ties of kinship and assisting Muslims. "Whoever relieves a believer of a worldly grief, Allah will relieve him of a grief on the Day of Resurrection, and whoever eases a difficulty for one in hardship, Allah will ease it for him in this world and the Hereafter." Furthermore, the most beloved of people to Allah are those most beneficial to people, and the most beloved of deeds to Allah is joy that you bring to a Muslim, or relieving him of a grief, or paying off his debt. Lending a dinar is more beloved than giving it in charity, because it returns and can be given in charity again, thus earning a double reward. "Whoever desires to have his provision expanded and his life prolonged, let him uphold his ties of kinship."