To see what tools your Busybox come with you should run it without options. Busybox is quite configurable so you should check documentation generated with the same configuration as your target. That web page is either very outdated or generated from some sample configuration.

Besides, ash also has $RANDOM so using AWK isn't really necessary:

echo $((RANDOM%firstnames_len))

Here is also a fix for Dave's suggestion:

i=1; for name in A B C; do echo "$((i++)): $name"; done

And if you use preincrement the i=1 is also unnecessary:

for name in A B C; do echo "$((++i)): $name"; done