What size disher makes "1 inch balls of dough?"

Well, a one-inch diameter sphere has a volume of 1.74 teaspoons, or 0.58 tablespoons.

It looks like the numbered sizes are in fractions of a quart, so if you could have any size you wanted, that'd be a a #110 disher. I assume that means you'll probably want a #100, which is 0.64 tablespoons, surely close enough for cookies. (From that same link, it looks like the sizes aren't that consistent to begin with, with examples often a few percent off.)

Note that this is based on the assumption that you care about volume, i.e. the amount of cookie dough per cookie. That seems right, but if you ever cared about the width of the scoop (something that doesn't spread, I guess) you'd need a larger scoop.

For general reading on scoops for cookies, this King Arthur blog post is good. They indeed suggest a #100 for smaller (2" to 2-1/4") cookies, where the recipe suggests a "teaspoonful" of dough - presumably a heaping teaspoon. For "tablespoonful" they suggest a #40 (4 teaspoons). They also mention using #30 and #16 for large cookies, but those don't correspond to common recipe instructions.

(The King Arthur scoops are also a great demonstration of the variation in size - they say their #40 is 4 teaspoons, when a #40 is nominally 4.8 tablespoons.)

For what it's worth, I took a list of standard sizes and did the same calculation as above to figure out what size sphere is equivalent to their nominal volume:

        Volume            Sphere diameter
        Tbsp    mL        in    cm
#4      16      236.59    3.02  7.67
#5      12.8    189.27    2.80  7.12
#6      10.7    158.22    2.64  6.70
#8      8       118.29    2.40  6.09
#10     6.4     94.64     2.23  5.65
#12     5.33    78.81     2.09  5.32
#16     4       59.15     1.90  4.83
#20     3.2     47.32     1.77  4.49
#24     2.67    39.48     1.66  4.22
#30     2.13    31.50     1.54  3.92
#40     1.6     23.66     1.40  3.56
#50     1.28    18.93     1.30  3.31
#60     1.07    15.82     1.22  3.11
#70     0.914   13.52     1.16  2.96
#100    0.64    9.46      1.03  2.62

Given the variability in scoop size across manufacturers, and the fact that you can scoop anywhere from level to heaping to full rounded spheres with the same scoop, this should probably just be a starting point.

I would use a 1-inch disher, in other words, a disher which is one inch across. A slightly heaping scoop with one of these will give you one inch balls of dough. If you fill the disher level each time you may be a tiny bit more consistent since you won't be eyeballing anything, but your cookie may also be slightly smaller than intended (and therefore bake perhaps just a little quicker).