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What merit did Batya, the daughter of Pharaoh have that she was the one to pick Moshe Rabbeinu from the waters?

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The Rogatchover explains that without her, the basket wouldn't have been in the water at all. The Egyptians worshipped the Nile and the canals branching from it, but the natural formations there (such as the reeds) wouldn't thereby become forbidden for use as an object of idolatry. So Yocheved placed the basket in the reeds. Then along came Basya to immerse in the water in order to convert to Judaism, and with that she "nullified" the river as an idol - so now Moshe's basket could enter it. (See footnote *ח there.)

ותשם בסוף על שפת היאור. ולא בתוך היאור, מחמת דהם היו עובדים לנילוס, רק דאין בו תפיסת ידי אדם, ועי׳ פסחים דף נ״ג סימן לנחלים קנים ע״ש מ״ש, ר״ל דנחל איתן למ״ד שלא נשתמש לעבר, אבל תוך היאור לא דהוה תפיסת ידי אדם, ב״מ דף ק״ג, ע״ב (ונ״מ) אריסא. רק כיון דבת פרעה הלכה לרחוץ ביטלה [הע״ז] שזה להתגייר ושוב באתה התבה בתוך היאור

The point, then, is that (according to this source, at least) the question is a nonstarter. Without Basya being there and immersing herself in the river, Moshe wouldn't have been in it in the first place; he would have remained among the reeds, and eventually Yocheved would have taken him back from there. You could say, I suppose, that her merit was that she was the one Egyptian at that time who was converting to Judaism and thus nullifying the river as an object of idolatry, but then her saving Moshe is a side effect of the free-willed choice she made to undertake that conversion.

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