I have heard that one of the prophets was mentioned 25 times in the Quran, but I can't find which one it is. I've heard this is a commonly taught Muslim tradition, and I think it is about Prophet Isa.
Isa (Jesus) and Adam are explicitly mentioned exactly 25 times in the Quran. This was found by searching “Jesus” at The world's most powerful digital tool for studying the Qur'an: its text, composition, and transmission and “Isa” and checking if any were parenthetical references (3 parenthetical references are found for Jesus, and 1 for Isa). This does not include the “generic” mentions, such as the references to “all the previous Prophets,” “that which was revealed beforehand,” and the like, as the other answers describe. Other references that refer to Jesus implicitly are not included, such as the references to the Messiah and/or son of Mary in Q 4:172, 5:17, 5:72, 75, 9:30–31. The parenthetical references are 4:159 for “Isa” and 3:3 & 43:57, 61 for “Jesus.” Note that this search mechanism looks through the Pickthall, Yusuf Ali, and Shakir translations, and I manually checked the (transcribed) Arabic to confirm the presence of “Isa.”
The references that include “Jesus” are: 2:87, 136, 253; 3:3, 45, 52, 55, 59, 84; 4:157, 163, 171; 5:46, 78, 110, 112, 114, 116; 6:85; 19:34; 33:7; 42:13; 43:57, 61, 63; 57:27; 61:6, 14.
This list includes 25 total references. This list includes references that refer to Jesus as the Son of Mary, Messiah, and as the Word of God (see surah 3:39, 45; 4:171 and compare John 1:1, 14).
Using the same searching method as described above (which can be confirmed by using the Quranic Arabic Corpus search mechanism): Moses is mentioned over 100 times. Abraham is mentioned over 50 times. Noah is mentioned over 30 times. Adam is mentioned 25 times exactly. Jacob is mentioned about 15 times. Jonah is mentioned exactly 4 times. Muhammad is explicitly mentioned only 4 times (3:144, 33:40, 47:02, 48:29). This Quora question Which prophet is mentioned most in the Quran? details more information on each one. Another helpful list: List of characters and names mentioned in the Quran - Wikipedia.
Conclusion: Jesus and Adam are both mentioned exactly 25 times in the Quran.
This is also interesting because the Quran and the Bible teach a connection between Adam and Jesus. The Bible teaches that Jesus is the Second/Final Adam. See 1 Corinthians 15:45-49 below and also see Romans 5:17.
“The first man Adam became a living being; the last Adam, a life-giving spirit. The spiritual did not come first, but the natural, and after that the spiritual. The first man was of the dust of the earth, the second man from heaven. As was the earthly man, so are those who are of the earth; and as is the man from heaven, so also are those who are of heaven. And just as we have borne the likeness of the earthly man, so shall we bear the likeness of the man from heaven” (1 Corinthians 15:45-49).
The Quran teaches that the creation of both Adam and Jesus occurred without human origin. They came from the Word of God, rather than from human beings.
Q (3:59): Indeed, the example of Jesus to Allah is like that of Adam. He created Him from dust; then He said to him, "Be," and he was.