Can you ride a storm to save fuel?
Not storms, but there is a concept called &quot;Pressure Pattern Flying&quot; where you plan routing to stay in favourable circulation around Highs and Lows, to the extent that deviations to follow the circulation flow and stay in tailwinds can get you there sooner than going straight.  To take advantage of this you need to be going fast enough and with the range to span pressure regions. Although Lindbergh talked up this sort of thing in the 30s in his own flying, it's generally something exploited by airliners on transoceanic flying.
And in fact the North Atlantic Track system adjusts the tracks northerly or southerly daily to maximize or minimize the effects of winds around pressure systems, especially focused on jetstreams, which form at frontal boundaries at the tropopause. It might be worth a significant deviation from straight line point-to-point flight to ride downstream in a 100kt 10 mile wide jet that's going your way at 31000 ft (or whatever level the tropopause is at for a given region and time of year).
As far as storm systems go, nothing good happens near one.  You stay away from them regardless of the size of the aircraft, 20 miles being the convention, and ideally on the upwind side.  One hazard of passing by a thunderstorm cell on the downwind side, even if well clear of it, is running into hail falling in the clear air after it shot out the top of the cell.