Ist and Bul are both common Turkish verb roots (istmak and bulmak, to want and to find)

does the name of the city signify some combination of these two ideas?

According to Wikipedia, the name Istanbul comes from the greek "is-tan-polís" (to go to the city).

it is derived from the greek terms "Ης την Πόλη" (is tin poli) or "στην Πόλη" (stin poli or stimboli) which means "in the city" or "to the city".

I thot it Istanbul ...shortened in such a way from the original name Constantinople (however its spelled.) that some meaning of the word was kept. Is there any significance?

Canstantinople from the latin "Constantine Polis", was Constantines City. The Roman Emporer than moved his capital from Rome to Byzantium, the original greek name of the city. (If that history isn't exact I apologize, maybe it was an Eastern capitol of Rome)

But It could be a contraction of "Konstantiopolis".

All your answers have the "a ring of truth". But I wonder why the Turks would have based its name on Greek words. Lets see what the others say.

"constantinople" (city of constantine) is the english version, "konstantinoupolis" is the greek version, "constantinopolis" is the latin version.

you can have a look at these sites for a few explanations:
http://en.wikipedia.org/wiki/Names_of_Istanbul
http://snailstales.blogspot.com/2006/05/anatolian-toponymy-2-from.html

apparently the ottomans heard the greeks say "is tin poli" (to the city) often and started using it as a name. but there were many many other names used for the city as well.

from the wikipedia link...
"After the creation of the Turkish Republic in 1923, the various alternative names besides İstanbul became obsolete in Turkish. In an edict of March 28, 1930, the Turkish authorities officially requested foreigners to cease referring to the city with their traditional non-Turkish names (such as Constantinople) and to adopt İstanbul as the sole name also in the foreign languages."

OK, I read the references. Thank you!

I remember that it was İslam + bol = (there are) many muslims.

After Byzantium, Muslim people became majority.

http://en.wikipedia.org/wiki/Names_of_Istanbul#.C4.B0slambol

Names of İstanbul

I don't know if there is another city in the world called with different names as many as İstanbul. Thanks to its rich
history.

From Wikipeida:

İstanbul is called with different names in those languages.

* Greek: Vizantion, Stampoli
* Latin: Bizantium, Antoninya, Alma Roma, Nova Roma
* Rumca/Old Greek?: Konstantinopolis, Istinpolin, Megali Polis, Kalipolis
* Slavic: Çargrad, Konstantingrad
* Hebrew:Kushta
* Viking: Miklagord
* Armenian: Vizant, Stimbol, Esdambol, Eskomboli
* Arabic: Bizantiya, el-Mahsura, Kustantina el-uzma
* Seldjuk Language: Konstantiniyye, Mahrusa-i Konstantiniyye, Stambul
* Ottoman Language: Dersaadet, Deraliyye, Mahrusa-i Saltanat, Istanbul, Islambol, Darü's-saltanat-ı Aliyye, Asitane-i Aliyye, Darü'l-Hilafetü 'l Aliye, Payitaht-ı Saltanat, Dergâh-ı Mualla, Südde-i Saadet, Kostantiniyye ( قسطنطينيه )

It is written differenlty in those languages now.
* Spanish : Estambul
* Hungarian : Isztambul
* Litvanian : Stambulas
* Lethuanian : Stambula
* Albanian : Stambolli
* Galic : Iostanbúl
* Loglanca: Konstantinupol/No idea where this language is being spoken
* Lazca/Greek dialect ???: Poli