In your question, you state:
Prophets mentioned in the Jewish and Christian holy books can be for example assumed to be known to common people.
Of the 25 prophets mentioned in the link you supply, 22 are mentioned in the Hebrew Bible (which is called the Old Testament by Christians), and 24 are mentioned in the Christian Scriptures (Yahya and 'Isa, known as John the Baptist and Jesus to Christians, postdate the Hebrew canon by several centuries). The 25th, of course, is the Prophet himself, who postdates the final works of the Christian Bible by about 500 years.
Christians and Jews see these prophets as some of the most important characters in the story of the redemption of Israel. They are listed approximately chronologically, and each contributes to the "grand narrative of God's love for his people."
I am going to recount the arc of the Old Testament narrative in broad strokes. At a high level there is:
- The Creation to the selection of Abraham to be the blessed one (Where the World began)
- The initial promise to Abraham that takes the people into slavery in Egypt
- Rescue from Egypt to the founding of the Kingdom of Isarael
- The destruction of Israel to their return from exile in Babylon / Persia / et al.
You will also often hear Christians use the words "Creation / Fall / Redemption" to describe this narrative.
- Adam (Adam) was the first man whom God created
- Enoch (Idris) "walked with God," and because God loved him, took him off the earth / Enoch never died
- Noah (Nuh) - and the flood - God saw him as an upright man in an otherwise reprobate culture
- Eber (Hud) - a minor figure in the Old Testament, but associated with the Tower of Babel incident in Jewish tradition
- Selah (Saleh) - was the last of Judah's sons, after a "special incident" This is a minor incident in Hebrew Scripture, but because of its age has special status. Some Christians also see this as indicative of God's nature - especially through the actions of Selah's wife, Tamar.
- Abraham (Ibrahim) - Genesis 12:1-3, God promised to bless him and to bless all nations through him
- Lot (Lut) - Abraham's nephew, and the one who attended him on his journey
- Ishmael & Isaac (Isma'il & Ishaq) - eldest son of Abraham. Christians / Jews and Muslims have a radically different understanding of the story, but in Muslim tradition, Ishmael is the one to be sacrificed
- Jacob (Yaqub) - the father of Israel, God's chosen nation to Jews and Christians
- Joseph (Yusuf) - God's vehicle for preserving his nation in Egypt
The above prophets constitute just about every major male character in the narrative of Genesis.
From Egypt to the Kingdom
- Job (Ayyub) - a cosmic story of a man who loved God, even when Satan tormented him. His chronological position is unknown
- Jethro (Shu'aib) Moses' non-Jewish Father-in-law, sent by God to teach Moses how to care for his people
- Moses (Musa) - the man whom God used to deliver his people and through whom the covenant was made
- Aaron (Harun) - Moses' brother whom God sent to speak to Pharaoh when Moses was afraid, and who was God's first priest
At this point, you have covered the remainder of the Torah, and mentioned all the "good guys" in the story. The Biblical narrative continues in Joshua & Judges, and recounts the story of the people through whom God gave the land of Canaan to the Jews. As this is for the benefit of Israel, I can see why the Quran would pass it over
The Kingdom of Israel
- David (Dawud) - the "man after God's own heart." Fought Goliath, wrote the Psalms, and was the King to whom God promised that he would establish his seed for ever and ever.
- Solomon (Sulayman) - David's son and the King who asked God not for wealth or power, but wisdom - the model under which God intended for people to live.
- Elijah (Ilyas) - the greatest of the Prophets (for Jews and Christians), through whom God spoke to wicked kings who had corrupted his nation
- Elisha (Alyasa') - Elijah's successor who inherited "a double portion of his spirit" and continued Elijah's work
- Jonah (Yunus) - the well-known, reluctant Jewish prophet to Ninevah who spoke God's truth, and testified to God's mercy (in spite of what he wanted!)
If I had to pick the 5 most important characters from the Kingdom time, it would be hard to compete with this list. (I might add Samuel, but again, as Israel's chief priest, I can see why he might have been ommitted)
Exile in Babylon
- Ezekiel (Dhul-Kifl) - The (crazy?) prophet to whom God sent the vision that Israel would be restored (Ezekiel 36)
- Zechariah (Zakariyya) - Another apocopalyptic prophet
Here, there are lots and lots of choices, and the two that are named are not insignificant (indeed, each has their own book!). Might I have chosen Isaiah, Jeremiah, or Daniel? Sure. But there is nothing that would have said these are "wrong" as being the highlights of the Exile either. It wasn't really a happy time for anyone!
The New Testament
After years of exile, Christians believe that God did restore his heavenly Kingdom, through Jesus ('Isa), and made good on the whole Creation / Fall / Redemption theme. His coming was announced by John the Baptist (Yahya).
The point of recounting the whole Christian / Jewish narrative is this - Just about any person in the Christian East (which Syria, Palestine, and even parts of the Arabian peninsula were) would have been familiar with the broad outline of this story. These 22 - 24 men do a really good job of summarizing the entire story, and as such, would have been well known.
Is there something mystical about 25? Not that I know of. Could more have been included? Less? Sure - but the list as it exists is pretty comprehensive in telling the entire tale of God's love for his people.