To rate "all-aroundness," I used a geometric mean. For example, if a player was 10 runs better than average in all three stats, I multiplied 10*10*10 = 1000. Then I took the cube root of that, which gives us 10. The geometric mean rewards balance, so a guy with 20 in all three stats is better than someone who had 100 in batting and just 1 in each of the other two.
Here is the top 25. It seems heavily weighted to 1980-2001 (15 cases) and it seems that base running usually had to be pretty good. There were only two seasons from 1911-1960. There were 8 cases from 1996-2001, by 8 different players.
Click here to see the complete rankings. After the top 50 or so, you might start to see duplicate entries. That is because as I called up these lists, some cases appeared on both of them and then I merged them all together and sorted them to get the ranking.
Seasons with 44.1+ batting runs, 0.1+ fielding runs, 0.1+ base running runs (197 cases)
Seasons with 0.1+ batting runs, 17.9+ fielding runs, 0.1+ base running runs (198 cases)
Seasons with 0.1+ batting runs, 0.1+ fielding runs, 5.8+ base running runs (190 cases)
Seasons with 30+ batting runs, 3+ fielding runs, 3+ base running runs (161 cases)