The 23-year-old's new deal means he is now under contract with the Championship club until 2019.
Bonham has played seven times for the Bees, making his Championship debut against Barnsley in April.
Brentford boss Dean Smith said Bonham "needs to play matches" to fulfil his "great potential".
Brentford goalkeeper Jack Bonham has signed a one-year contract extension with the club and then joined League Two Carlisle on loan until January.