How authentic is the Hadith regarding a woman saying to her son: "He was killed while fighting as a martyr," to which the Prophet, peace and blessings be upon him, replied: "And what makes you so sure he is a martyr?"

Perhaps the questioner has confused two hadiths:

The first: The hadith of Umm al-Ala al-Ansariyyah upon the death of Uthman ibn Mazun, when she said: "May Allah have mercy on you, Abu al-Sa'ib, for I bear witness that Allah has honored you." The Prophet, peace and blessings be upon him, said to her: "How do you know that Allah has honored him?" Then he said: "As for him, certainty has come to him, and by Allah, I hope for good for him. By Allah, I do not know—and I am the Messenger of Allah—what will be done with me?"

And the second: The hadith of Umm Harithah al-Ansariyyah, who asked about her son who was killed on the Day of Badr. She said: "O Messenger of Allah, you know the place Harithah holds in my heart. If he is in Paradise, I will not weep for him; otherwise, you will see what I do." He said to her: "May you be bereaved! Is it only one Paradise? Indeed, there are many Paradises, and he is in al-Firdaws al-Ala (the highest Paradise)."