Answer: D

Small Hint:

The ones digit of $n^2$ depends only on the ones digit of $n,$ so the list repeats every $10$ terms

Big Hint:

Sum one full block of $10,$ multiply by the number of blocks, then add the leftover terms

Solution:

The ones digits of $1^2, 2^2, \ldots, 10^2$ are $1, 4, 9, 6, 5, 6, 9, 4, 1, 0,$ which sum to $45.$ The $2025$ terms contain $202$ full blocks ($2020$ terms) plus $5$ more with digits $1, 4, 9, 6, 5$ summing to $25.$ The total is $202 \cdot 45 + 25 = 9090 + 25$ $= 9115.$

Thus, the correct answer is D.