Your likely to get a basket full of recommendations but I'll start out with these.
Both Mossberg and Remington make a youth model gun in 20 ga. The Model 870 Remington, and the Model 500 Mossberg. Winchester also made the youth model 1300 but they stopped producing the 1300 this past spring.
These are all pump guns and 3 inch chambered. with shorter stocks and typcally shorter tubes as well. Both the stocks and barrels can be replaced with full size for future use.
There is also the youth Model 1100 by Remington which is a 20ga. auto.
If wanting to go up to a 12ga. you would be much more limited, but shorter stocks can be had for many of the 12 ga. guns as well - just generaly not as part of the factory package.
At 11 I would personaly start him/her with a 20ga youth model. Even if going to a 12ga. in a few years, that 20 ga. will serve him/her a lifetime for upland game and other uses where a 12ga. isn't needed.