The Obis (kings, effectively) of Onitsha claim descent from Edo royalty (Kingdom of Benin), via a 16th-century schism that sent one prince eastward though the cause is disputed. The polity itself was a diverse fusion although Igbo-speaking and identifying today, so I'm not sure I can quite go with the modern ethnic identifier you're using for a son of the Obi at that distant time. That said, there's an awful lot about Onitsha we don't know for such an early period--it had a core area of a few major towns, and later spread out to exert hegemony over a slightly greater area in the vicinity of the former Igala kingdom.

I will go have a look for any direct accounts of Onitsha governance and authority from the 1600s but because it became really prominent only in the late 18th century (when direct traffic that far up the Niger became more voluminous) there may be little. The position of red-capped first rank chiefs and the council system is a modern regularization of earlier council systems, so we're even at a bit of a loss to figure out just how absolute an Obi's power was. It deviated radically from the decentralized and almost republican operation of Igbo societies, but it also wasn't quite the sophisticated system of hierarchy of the Obas of Benin. In any case, members of the royal household had distinct positions and webs of authority and subservience that helped to hold the state itself together.

So in a general sense, the answer would be "monetarily or materially, not wildly so; but in terms of patronage and social position which gave power over people and within networks, enormously, yet constrained under the umbrella of the Obi's own power." The position of prince would also vary depending on whether there was a system of heir-presumption or election to succession (both existed in the region), and whether the basic standing of "prince" persisted for brothers of the new Obi who could have a very different social standing and set of powers than they did as minors in the Obi's household. None would have possessed the independence to deal separately with foreigners, because the state was not falling apart; one thing that the enduring polities in that region almost always shared was a great deal of control over the interface between their societies and outside traders, made easier by their relatively small size.

For the specifics of how the state worked, I'm going to dig into my copy of Groundwork of Nigerian History, Ryder's Benin and the Europeans, and a couple of other texts I have to see if anyone spent any time with the mechanics of that particular state and its general sources of wealth in the 1600s that might provide a more precise answer than the one I must extrapolate here.