Vines has
oikonomos (οἰκονόμος, 3623) primarily denoted "the manager of a household or estate" (oikos, "a house," nemo, "to arrange"), "a steward" (such were usually slaves or freedmen), ...; in Rom. 16:23, the "treasurer" of a city (see CHAMBERLAIN); it is used metaphorically, in the wider sense, of a "steward" in general...
oikonomia (οἰκονομία, 3622) primarily signifies "the management of a household or of household affairs" (oikos, "a house," nomos, "a law"); then the management or administration of the property of others, and so "a stewardship"...
The differing etymological suggestions are curious, but that's an aside.
The key idea, then, is that Paul has become a servant entrusted by God with overseeing other servants, but that's hard to gloss in a post-feudal culture. We could try head butler, but I'm not sure that most people would understand what a butler's job really is. It might be possible to gloss it by analogy with Joseph's role either in Potifar's household or as second-in-command to Pharoah. Maybe the solution is to take a dynamic approach and look for a similar rôle in a modern business setting: I have been appointed COO by God... Or we could follow the NIV's approach and take a military analogy: the commission God gave me.
The responsibility God gave me captures a lot, but leaves the reader to work out what that responsibility actually was. In the context of a word ministry and following through on the imagery of a servant who oversees other servants to ensure that the whole estate works well, it's something like the responsibility God gave me to ensure that you fulfil his purposes for you.