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I'm a little bit familiar with how の is to indicate possession but this just confuses me. Does it translate to "Grant my wish for a family"?
願いの叶う家 = 願いが叶う家 as @choco said. A general rule is that if the construct is X の Y Z where X is a pronoun or noun, Y is a verb, and Z is a noun, then の is really が.
X の Y Z
For more on why, see the link above.
It translates to
"A house that makes wishes come true."
Not sure what the context is of the sentence, with that alone it seems to be the above.