How can I generate new variable names on the fly in a shell script?

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You need to utilize Variable Indirection:

SAMPLE1='1-first.with.custom.name'
SAMPLE2='2-second.with.custom.name'

for (( i = 1; i <= 2; i++ ))
do
   var="SAMPLE$i"
   echo ${!var}
done

From the Bash man page, under ‘Parameter Expansion’:

“If the first character of parameter is an exclamation point (!), a
level of variable indirection is introduced. Bash uses the value of
the variable formed from the rest of parameter as the name of the
variable; this variable is then expanded and that value is used in the
rest of the substitution, rather than the value of parameter itself.
This is known as indirect expansion.”

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