We LOVE LOVE a day with some heavy rain. The sweet spot seems to be mid afternoon. Why? By then, people have been there all day and will leave in droves. And the rain has never lasted all that long. Just make sure you take a poncho or two (the disposable dollar kind), because Disney is delighted to bring out their plastic raincoats that are so expensive is feels like price gouging.
Either find an indoor retreat, or walk around with your rain gear, and watch everyone leave. Then the rain will go away (it always does) and enjoy your leisurely late afternoon and evening at the park.