I came across this sentence:
Er ist mein zu Hause geworden.
First off, what does the preposition zu add to the meaning of the verb werden?
Secondly, Why is mein left by itself here? Can it be replaced by mir and still convey the same meaning?
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You probably have a typo or grammatical error in that sentence. I'd assume it meant to read
Er ist mein Zuhause geworden.
which would translate as something like
He became my home.
"Zuhause" is a "Zusammenrückung" (compound or composition) of "zu Hause". An English example of the same principle would be something like "bystander" (from "someone who stands by") or "onlooker". As those, "Zuhause" is a noun. You can think of it as a short form of "wo ich zu Hause bin" ("where I am at home"), which is my "Zuhause" (my "home").
Regarding your second question, "mein" could be replaced by "mir". That would be a different grammatical construction. "mein" denotes a genetive, as in "my home". With "mir", the sentence would rather read
Er ist mir ein Zuhause geworden.
and translate as
He became a home to me.
Grammatically, the genetive is replaced by a dative. And as far as the meaning is concerned, I'd say it's somewhat weaker. "Er ist mein Zuhause geworden" has a connotation of "He became my one and only home", while "Er ist mir ein Zuhause geworden" means more something like "he's a valued home right now, but not necessarily the only one, and it can change down the line".
(h/t to @Beta and @see-sharper who mentioned some of this already in the comments to your question)
The fixed adverbial phrase "zu Hause" (at home) is in the classic spelling version of what can also be written as a one-word adverb 'zuhause' (per orthographic reform of 2006, but who does that, anyway…). The noun version would be '(das) Zuhause', meaning: the place one is/feels at home.
Er ist mein "zu Hause" geworden.
He / It has become my (being at) home.
(It's "he", if the pronoun stands for a male person, but "it", if "er" stands for a masculine noun.)
While it technically should be the noun 'Zuhause', the author may have chosen to write "zu Hause" deliberately (it's allowed for 'zu Hause') to emphasise it as a key phrase, so as to express the quality of being at home (that is: having found a place to really call 'my home') as an important emotional state.
┏━━SUBJECT━━┓ ┏━━AUXILIARY━━┓ ┏━━━━━━━OBJECT━━━━━━━┓ ┏━━━MAIN VERB━━━━┓ Er ist mein Zuhause geworden. └─PRONOUN─┘ └─AUXILIARY─┘ └─NOMINATIVE NOUN PHRASE─┘ └─COPULAR VERB─┘ masculine present tense Prädikativ past participle
The sentence is in the present perfect tense (Perfekt): "ist" = auxiliary verb 'sein' (to be) in the singular 3rd person, present tense + "geworden" = past participle (2. Partizip / Partizip Perfekt) of 'werden' (to become).
About the case of the object: The phrase »mein "zu Hause" / Zuhause« is in the nominative case – this is called Gleichsetzungsnominativ or prädikativer Nominativ – the nominative of equation or the predicate nominative: An object of one of the copular verbs (Kopularverben) sein, werden, bleiben is equated with the subject (Er), and that puts the object phrase in the nominative case (Er ist wer oder was geworden? Mein "zu Hause".)