Richardson hit an unbeaten 83 off 91 balls, and shared a stand of 121 with skipper Keaton Jennings (79).
After an earlier shower, the rain set in with Durham 194-2 after 34 overs.
Derbyshire were eventually set a revised target of 243 from 34 but the weather closed in again before the teams could return to the middle.
Stephen Cook and Jennings had launched Durham's innings by putting on 47 before the latter played across the line to the final ball of the eighth over, bowled by Ben Cotton, and was lbw for 18.
Jennings reached his half-century off 49 balls, with the help of seven fours, but was stumped in the 29th over after going down the pitch to spinner Jeevan Mendis.
Durham batsman Michael Richardson made his highest 50-over score before rain forced the abandonment of their One-Day Cup opener against Derbyshire.