Y in this case is replacing the use of “un ordinateur”, effectively making the statement less verbose.
For example, you could phrase it as:
“J’ai un ordinateur et des jeux mais c’est mon petit frère qui joue toujours a l’ordinateur”
and it would make just as much sense.
It would be the same in English as contracting the sentence:
“I have a computer and some games, but my little brother plays on the computer all the time”
“I have a computer and some games, but my little brother plays on it all the time.”