According to me, the best way is: "Ha detto che non è riuscito a connettersi [ad Internet] da casa".
We normally use "connettere" (to connect) referring to the internet, so you might hear many people omitting the object (Internet) in such sentences.
There is another expression which is "collegarsi ad Internet/alla rete". Collegare is not a verb so strictly linked to the Net, so you may want to specify the subject, unless it is obvious to your listener from the context.
This said, despite being less technical, some people use also "andare su internet", which is correct, too!