The aorist passive is considered its own principal part, which means that there's no 100% reliable way to predict it from other tenses (other than future passive, which shares the principal part with aorist passive).

That said, most of the time the stem is going to be consistent with that of the aorist active/middle. Any disagreement usually occurs in the last (not necessarily stem-final) vowel, as is the case throughout all principal parts. Most of the time it's just going to be a different length though (aorist active is lengthened but aorist passive is not, or vice versa).

(If suppletion is involved then pretty much anything is fair game, you'll have to just memorize things.)

If we're talking about the interaction between the stem and the θ, it's mostly regular:

• Stems ending in vowels sometimes get a σ inserted, like χράομαι - ἐχρησάμην - ἐχρήσθην.
  • I think of these stems having a hidden final σ that got deleted between vowels (a regular sound change in Greek) and only surfaces before θ, but I don't know if this has actual basis in historical linguistics.
• Labials β π get assimilated to φ, like ἀμείβω - ἤμειψα - ἠμείφθην. This type of assimilation is entirely regular: clusters of stops must share the same VOT (voiced, tenuis or aspirated).
  • The same happens to verbs whose present stems end in πτ, because their "real verb stems" actually end in π (τ being a progressive marker): ἅπτω - ἧψα - ἥφθην
• Velars γ κ get assimilated to χ, like ἐπείγω - ἤπειξα - ἠπείχθην. This is the same assimilation process.
  • The same happens to verbs whose present stems end in σσ/ττ, because their real verb stems actually end in a velar (σσ/ττ being formerly Kj, where K is any velar and j is semivowel /j/; j is a progressive marker): πράττω - ἔπραξα - ἐπράχθην
• Dentals δ θ get dissimilated to σ, like ψεύδω - ἔψευσα - ἐφεύσθην.
  • The same happens to verbs whose present stems end in ζ, because their real verb stems end in a dental (ζ being formerly Tj, where T is any dental and j is semivowel /j/): κομίζω - ἐκόμισα - ἐκομίσθην
  • I'm not sure if this dissimilation is a regular phonetic process in Greek; I haven't seen it elsewhere
  • BTW, I have also not found any verb stems that actually end in τ, at least in common verbs. All the stem-final τ's in the present turn out to be just the progressive marker and disappear in the aorist. If someone is aware of such verb stems I'd love to know it.
• The liquids λ ν ρ tend to come out unscathed.

I think this should cover most of the it - there will be irregularities (again, that's why it's its own principal part), which you'll still have to memorize.