I tend to think the texts are red herrings and that a family member did it, but if not, perhaps the killer seeks revenge for something that happened in Hungary. Was the victim by chance Roma?
On the photo she does not look Roma at all... and Hungary does not have that many (unlike Romania).
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I have no clue what this is but (1) this is not a random murder and and not a robbery (2) this is a very low crime neighborhood with very functional police department and thus lots of publicity and there will be a lot of attention to solving it too (3) may be worthwhile checking the profession of the husband — perhaps he is the one responsible.