The Presidency of the Sanhedrin was a Shadow Monarchy

Mishnah Hagigah 2:2  lists the Nsei'im who preceded Hillel:

יוסי בן יועזר אומר שלא לסמוך, יוסי בן יוחנן אומר לסמוך.
יהושע בן פרחיה אומר שלא לסמוך, נתאי הארבלי אומר לסמוך.
יהודה בן טבאי אומר שלא לסמוך, שמעון בן שטח אומר לסמוך.
שמעיה אומר לסמוך. אבטליון אומר שלא לסמוך.
הלל ומנחם לא נחלקו.
יצא מנחם, נכנס שמאי. שמאי אומר שלא לסמוך, הלל אומר לסמוך.
הראשונים היו נשיאים, ושניים להם אבות בית דין.

Translation from Sefaria:

Yose ben Yoezer says not to lean hands [on the Chagigah sacrifice]; Yose ben Yohanan says to lean hands. Yehoshua ben Perahia says not to lean hands; Nitai the Arbelite says to lean hands. Yehuda ben Tavai says not to lean hands; Shimon ben Shetach says to lean hands. Shemaya says to lean hands; Avtaliyon says not to lean hands. Hillel and Menahem did not disagree. Menahem left and Shammai entered. Shammai says not to lean hands; Hillel says to lean hands. The first [of each pair] was the nasi [head of the Sanhedrin] and the second [of each pair] was the av beit din [vice-head of the Sandhedrin].

So the first five Nesi'im were:

• Yose ben Yoezer
• Yehoshua ben Perahia
• Yehuda ben Tavai
• Shemaya
• Hillel

The Bavli (Shabbat 15a) lists the three Nesi'im who followed Hillel:

הלל ושמעון גמליאל ושמעון נהגו נשיאותן (לפני) הבית מאה שנה 

Translation (including interpolations that are very convenient for my argument) from Sefaria:

Hillel, and his son Shimon, and his grandson Gamliel, and his great-grandson Shimon filled their position of Nasi before the House, while the Temple was standing, for a hundred years

So before Hillel the position of Nasi was not hereditary and following Hillel the position of Nasi became hereditary. That position remained hereditary, except for a short period after the destruction of the Temple while Rabbi Yohanan ben Zakkai filled in while Rabban Gamliel of Yavneh was growing up, and except for a brief time when the position was shared by Rabban Gamliel of Yavneh and Rabbi Elazar ben Azaryah, until the position was abolished by Theodosius II in 425 CE. What caused the change?

Hillel died in 10 CE (Wikipedia; GENi; Encyclopedia.com) or in 8 CE (Chabad) or some time between 10 CE and 20 CE (New World Encyclopedia) . (That's probably enough Googling of "Hillel the Elder" to make my point.)

In 6 CE, Augustus Caesar replaced Herod Archelaus, the last (nominally) Jewish ruler of Judea, with a Roman governor.

Originally, the Nasi had to merit his position by being one of the leading scholars of his generation. What could have motivated the Sanhedrin to change its criterion?

I propose that the motivation was political. I propose that the Sanhedrin made the position of Nasi hereditary in anticipation of turning the family of the Nasi into a royal dynasty of Judea when the Roman Empire falls at some unknown date in the future. This change came, of course, with the downside of the Nasi not necessarily being a great scholar. Rabban Gamliel I (the grandson of Hillel) and Rabbi Yehuda HaNasi arguably were the top scholars of their generation. Rabban Gamliel of Yavneh was outclassed at least by Rabbis Eliezer, Yehoshua and Akiva, which is what led to the rebellion that forced Rabban Gamliel of Yavneh to share his position with Rabbi Elazar ben Azaryah, as described in Brakhot 27b-28a. It also (IMHO) led to Rabbi Meir's failed coup d'etat, as described in Horayot 13b-14a, whose purpose was to restore election of the Nasi based on merit.











The following is Wikipedia's list of Presidents (נשיאים) of the Sanhedrin:

Yose ben Yoezer                     170 BCE - 140 BCE
Joshua ben Perachyah            140 BCE - 100 BCE
Simeon ben Shetach               100 BCE - 60 BCE
Sh'maya                                   65 BCE - c. 31 BCE
Hillel the Elder                        c. 31 BCE - 9 CE
Shimon ben Hillel                   9 CE
Shammai                                 9 CE - 30 CE
Rabban Gamaliel the Elder    30 CE - 50 CE
Rabban Shimon ben Gamliel 50 CE - 70 CE
Rabban Yohanan Ben Zakai  70 CE - 80 CE
Rabban Gamaliel II of Yavne 80 CE - 118 CE
Rabbi Elazar ben Azariah       118 CE - 120 CE
Interregnum (Bar Kochba revolt)
Rabban Shimon ben Gamliel II 142 CE - 165 CE




see also https://www.yeshiva.org.il/wiki/index.php?title=%D7%A0%D7%A9%D7%99%D7%90_%D7%94%D7%A1%D7%A0%D7%94%D7%93%D7%A8%D7%99%D7%9F#.D7.A8.D7.A9.D7.99.D7.9E.D7.AA_.D7.A0.D7.A9.D7.99.D7.90.D7.99_.D7.94.D7.A1.D7.A0.D7.94.D7.93.D7.A8.D7.99.D7.9F_.D7.91.D7.AA.D7.A7.D7.95.D7.A4.D7.AA_.D7.91.D7.99.D7.AA_.D7.A9.D7.A0.D7.99_.D7.95.D7.90.D7.97.D7.A8.D7.99.D7.94
(נשיא הסנהדרין)