Answer: 192

Small Hint:

The units digit forces the number to have the form $10a+2$

Big Hint:

Reduce $(10a+2)^3\equiv888\pmod{1000}$ first modulo $125$

Solution:

Write the number as $10a+2.$ Expanding the required congruence modulo $125$ and dividing by $5$ gives $$20a^2-a-1\equiv0\pmod{25}.$$ Reducing first modulo $5$ gives $a\equiv4\pmod5;$ substituting $a=4+5b$ then gives $b\equiv3\pmod5.$ Thus $a\equiv19\pmod{25},$ and the smallest candidate is $10(19)+2=192.$ It is even, so its cube is also $0\pmod8,$ and indeed $192^3=7{,}077{,}888.$