。
She wished that cute dog were hers.
Literal
She [topic-は] [that cute dog [subject-が] herself's-thing if-were-good [I wish-のに]] [quotative-と] thought.
~たらいいのに is the canonical Japanese frame for wistful 'I wish X were the case' — built from the conditional ~たら ('if'), いい ('good'), and のに ('but / even though'), it amounts to 'if it were that way, it would be good — but it isn't'. The のに carries the regret; remove it and the sentence flattens to a neutral 'it'd be nice if'. 自分のもの — literally 'her own thing' — uses もの as a generic possessor noun where English just says 'hers'. The whole wish is bundled by quotative と into the matrix verb 思った, which is how Japanese reports inner thoughts: inner monologue gets 思う, spoken words get 言う.