Abarim Publications' online Biblical Greek Dictionary
οιομαι
The verb οιομαι (oiomai) means to figure or suppose (John 21:25, Philippians 1:16 and James 1:7 only). This verb describes a mental act but not one securely carried by reason or rational contemplation but rather by intuition. In the classics this verb is used often in the sense of to forebode or presage, not magically or prophetically but rather by informed suspicion or by a boding presenting itself within one's heart (Od.19.312). In that sense, our verb may be used colloquially in the sense of to think that... or be of the opinion that ... or suspect that ...
Our verb may be used to convey otherwise indeducible intentions: "I mean to go north." Or doubt: "I think so, maybe". Our verb may be used to smooth over too bodacious an assertion: "methinks it is like a weasel" rather than, "now that's a weasel!" Or it may help introduce an otherwise taxing opinion ("I think we should go back"), or something one is not too sure of ("I think that might work..."). It may add force to consternation: "you can't possibly think that should go there!" Or to explain something that lacks alternative evidence: "I think it's down there." Or support otherwise inexplicable motivation: "I think it my duty to point out that further action is unwarranted."
This verb stems from a widely attested Proto-Indo-European root "hew-" to perceive, which left surprisingly few traces in Greek: another example is αισθανομαι (aisthanomai), to perceive or understand, which became in English the word "aesthetic". In Latin this PIE root resulted in the familiar word audio.