"habari" is a noun of the noun class, for which singular and plural form of the noun are equal. With very little exceptions "ya" and "za" have to be used to indicate genitive ("of") for this noun class. ya" is for singular, and "za" is for plural: habari (1) ya ... habari (>1) za ...

For other noun classes this genitive particle will change, e.g. jina (1) la ...

@Svea_DE is right, the "za" and "ya" could theoretically give a sense for how many "news-es" you're talking about. but if you're looking for a more practical answer, use whichever one you want. people say "habari za..." and "habari ya..." all the time.

the possessive here is implied, but you could use "kwako" if you wanted to make that even more clear (maybe if she had just asked about your home and you were turning it back to her). "habari za nyumbani kwako, dada?"

it wouldn't be "lako" because "nyumbani" is not in the ji/ma noun class. it would be "nyumbani kwako" because the "-ni" at the end of "nyumba" puts it in the locations noun class.