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According to Wiktionary, שלמה (pronounced /ʃloˈmo/ in Modern Hebrew) is the Hebrew version of Solomon. The pronunciation seems to follow reasonably well from the spelling, and as far as I can tell, it was spelled the same way in Biblical Hebrew.

The descendants look quite a bit different from the Hebrew pronunciation:

Ancient Greek: Σολομών (Solomṓn), Σαλομών (Salomṓn), Σαλωμών (Salōmṓn), Σολομῶν (Solomôn), Σολωμών (Solōmṓn)
 → Gothic: 𐍃𐌰𐌿𐌻𐌰𐌿𐌼𐍉𐌽 (saulaumōn)
 Greek: Σολομών (Solomón)
 → Latin: Solomon
 → English: Solomon
Classical Syriac: ܫܠܝܡܘܢ‎ (šlemūn)
 Arabic: سليمان‎ (Sulaymān)
 → English: Suleiman
 → Turkish: Süleyman

It seems reasonable to me that the /ʃ/ shifted to a /s/, and most of the languages added a vowel between the /s/ and the /l/, and that Syriac and Arabic changed the vowels.

However, I don't understand where the /n/ at the end came from. I'm not aware of any other Biblical names that ended in an open vowel in Hebrew and shifted toward having an n.

For the vowels, pay close attention to the nəquddoth (vowel dots)! Between the shin and the lamedh is a shəwa mark; sometimes this indicates an extra-short vowel, sometimes no vowel at all. But historically, shəwa was always pronounced (shəwa na) if it came after the first consonant in the word. So at the time of the Septuagint, the name was pronounced something like Shəlōmō.

When the Greeks tried to transcribe this word for the LXX, they ran into some difficulties. Ancient Greek didn't have [ʃ], only [s]. So they wrote the first letter as a plain sigma ("s").

Similarly, Ancient Greek didn't have any extra-short vowels corresponding to shəwa na, so they made do with the letters they had: in this case, it was sometimes transcribed with a short alpha ("a"), sometimes with an omicron ("o"). You'll also sometimes see shəwa transcribed with epsilon ("e"), depending on the word; this is closer to the modern pronunciation.

Finally, in Ancient Greek, it was extremely rare for names to end in -ō. It was much more common for them to end in -ōn (Glaucōn, Cleōn, Zēnōn). So the Greeks stuck a nu ("n") on the end to make the name look more like a name. This is also why English "Moses" ends in "s" when the Hebrew Mōshe (מֹשֶׁה) doesn't: -ē is a feminine ending in Greek, not a masculine one, so the translators changed it to the very common and masculine -ēs (Achillēs, Sōcratēs, Diogēnēs), and this version persisted.

One might expect the Romans to then remove this -n again, since the same "type" of name in Latin ended in plain -o (Cicero, Scipio, Piso). But the Latin Vulgate is generally more faithful to the Greek than the Hebrew, especially where names are concerned, so "Solomon" persisted.

P.S. Chromium fails beautifully when it tries to render the question title…