I did the exact same repair on a local kid's bike awhile back. The thickness of the cog was not an issue. The spacing was though, because in my case the one nine-speed cog set just a bit closer to the adjacent eight-speed cogs than the derailleur expected. As I recall, I set the high-limit to leave the chain centered on the new cog. Then I allowed a tad bit of slack in the cable that was taken up on the very first downshift. That slack (minimal) allowed the shifter to pull an eight-speed amount of cable while moving the chain across a nine-speed amount of distance.
Xenologer's point about a mismatch in wear between old chain and newer cog is a good one to consider as well.