(all sources available below)
The gemara attempts to address the seemingly unnecessary suffix (ה-ו) added to "וַתִּרְאֵ֣הוּ" (and explains that bat Pharaoh saw the Shechina, ה-ו being an allusion to HaShem's name). Other mefarshim ask why Moshe is described as both ילד and נער, the juxtaposition of synonymous descriptions seemingly superfluous. It seems that the pasuk is filled with extra, unnecessary letters and words. In fact, I would suggest that the phrase "אֶת־הַיֶּ֔לֶד וְהִנֵּה־" is unnecessary, and the pasuk would have the same meaning if is was simply written "וַתִּפְתַּח֙ וַתִּרְאֵ֣הוּ ... נַ֖עַר בֹּכֶ֑ה וַתַּחְמֹ֣ל עָלָ֔יו וַתֹּ֕אמֶר מִיַּלְדֵ֥י הָֽעִבְרִ֖ים זֶֽה׃" (and it would keep the gemara's explanation that bat Pharaoh saw the Shechina).
So what is the additional language adding?
Additionally, Or HaChaim hakadosh asks כי מנין ידעה השכינה מבית אביה ואבי אביה? how is it possible that bat Pharaoh would recognize the Shechina (he sarcastically asks whether it was possible she should recognize it from her father or grandfather, who clearly had no knowledge of the Shechina)? He answers that when bat Pharaoh's saw Moshe she saw a "big light" with him; one that HaShem provided to demonstrate to her that Moshe was special. But what was the "big light" that HaShem produced for her to see and how did it impress bat Pharoah?
Perhaps the impressive big light that bat Pharaoh saw is hinted to by the Targum Onkelos and Targum Yonatan and they simultaneously explain the necessity of the seemingly superfluous language. They each translate את־הילד והנה as "ית רביא והא".
Embedded in that translation is the following:
י = 10, an allusion to the aseret hadibrot that the bnai yisrael will receive through Moshe at har sinai
ת רביא = 613, an allusion to the 613 mitsvot that the bnai yisrael will receive from the Torah we receive from Moshe
ו-ה = and allusion to HaShem (see the gemara cited above)
א = 1 an allusion to the one-ness of HaShem (ו-ה א to be read together)
So putting all of this together, when bat Pharaoh saw Moshe, the "big light" she saw with him, the one that was intended to impress upon her how special Moshe was, was actually a nevua of the future accomplishments that Moshe achieve: she saw the aseret hadibrot and the Torah / taryag mitsvot, and she understood or even accepted the one-ness of HaShem. These were the "אור גדול" that Or HaChaim mentions. She saw, as hazal summarize in the gemara, the Shechina.
As an aside (perhaps another proof to the above?), the reference to the Shechina might further be rooted in the phrase את־הילד והנה, as follows: the gematria of הילד is 44, the same gematria as YKVK (26) Chai (18). Perhaps when the g'zeira against Jewish babies was issued, the mitzrim and some Jewish slaves thought that the Gd of the Jews who made Yosef so powerful, had abandoned the Jews. (See Nietzsche for a similar idea "Gd created the world, but now He is dead"; Aristotle held similar beliefs. We also see in Jewish history goyim assuming that HaShem has abandoned us when troubles befall us.) When bat Pharaoh saw Moshe and had her nevuah she realized that the Gd of the Jews is "still alive."
וַתִּפְתַּח֙ וַתִּרְאֵ֣הוּ אֶת־הַיֶּ֔לֶד וְהִנֵּה־נַ֖עַר בֹּכֶ֑ה וַתַּחְמֹ֣ל עָלָ֔יו וַתֹּ֕אמֶר מִיַּלְדֵ֥י הָֽעִבְרִ֖ים זֶֽה׃
ורז''ל דרשו (סוטה יב:) ותראהו ראתה שכינה עמו, והוא דרך דרש. וצריך לומר שלא הכירה השכינה היא, כי מנין ידעה השכינה מבית אביה ואבי אביה, אלא שראתה עמו אור גדול, וה' עשה כן כדי שתגדל מעלת הילד בעיניה. גם לטעם שדרשו ז''ל (זהר ח''ב י''ב:) בכתוב והנה נער בוכה שרמז על ישראל שהיו בוכים בגלות, לזה ויגל ה' את עיניה להראותה האור הסומך למשה.
וּפְתָחַת וְחָמַת יַת רַבְיָא וְהָא טַלְיָא בָּכֵי וְחָסַת עֲלוֹי וַאֲמָרַת מִן בְּנֵי יְהוּדָאֵי הוּא דֵין