Organisers had hoped to squeeze in the match before the expected weather front arrived, re-scheduling the final for mid-day on Sunday local time, but the rain started early and hardly relented.
Play was officially called off just before 6:00pm (3:00am AEST).
World number two Murray, who produced a stunning demolition of Andy Roddick in the semi-finals, is bidding to win the title for a second time following his triumph two years ago.
It is the third time the traditional Wimbledon warm-up event has been extended into Monday after similar weather problems in 1979 and 1987.
The 24-year-old Murray, who will again shoulder British hopes of a long-awaited men's singles champion at Wimbledon starting next week, killed time by playing table tennis against Tsonga and exchanging banter with fellow players on his Twitter feed.
In one message to world number two Novak Djokovic, en route to London via Luton airport, Murray suggested the Serb came to Queen's for a sing-along with the American doubles duo Bob and Mike Bryan.
Djokovic replied: "Another idea: Go out on court with water polo caps and speedo's...I will get a ball!"
Former world number one Serena Williams, given a wildcard at this week's Eastbourne tournament after nearly a year out with injury and illness, practised indoors with her sister Venus.
It was a similar story in Birmingham where the Edgbaston final between between Daniela Hantuchova and Sabine Lisicki was also postponed until Monday.
- ReutersTags: sport, tennis, england