I have probably spent the least time studying the oligarch story line but my impression is the BIP is a direct connection to the source.
The strange things that happened involving it are results of various items making contact with it.
the oracle said in Mx3 that Neo "touched the source" and that it should have killed him because his mind wasn't ready for it.
If we presume source is something like source code it then explains how the BIP can function in this way as using such code is how it could make a human into a program or turn a bluepill into a puddle
If this does give whoever manages to merge with it some kind of control over the source code then it would literally allow someone to control the machines and the matrix.
If we presume that Neo had it when he left the architect then it would explain what happened with the sentinels of the program interfaced with the source in order to self destruct the sentinels.
Causing Neo's unprepared mind to go a bit funny splitting Neo's body and mind allowing him to exist in the matrix (I'm guessing temporarily as Wright suggested) and if he had not been rescued would have faded away. (link does say how Neo was barely registering)
I think the reason people have been seeking the BIP are:
Halborn: as an oligarch has power over the machines but not direct control and as an outcast as he seemed to be was seeking to gain complete control over the machines and the matrix.
Carlyne: wanted to retrieve the BIP for the oligarchs who seem to be humans that are in charge to some extent.
Wright: with access to the BIP would likely be able to give Zion humans control of the matrix ie. Immunity from agents, ability to awaken humans that want to be and no longer fear the machines will destroy Zion/New Zion
Merovingian: doesn't seem aware of the programs true function so will study it to find its secrets as 3 of the most powerful human in recent matrix history have been after it.
Machines: because if the wrong person gets it they are up **** creek without a paddle!
Zion: Everyone else wants it so it must be worth nabbing!
These are my theories based on what i know so far and a bit of guess work.
PS. Rarebit.. if i am close can you give me some kind of a sign? i wont tell anyone im just interested to see how wrong i end up being.