The following is the reasonable verdict, I believe, given no more context, as said by Steven's comment and harold's comment.
The maximum (decimal) integer that can be stored in memory of 8-bit word processor computer depends on the context, that is, whether we are talking about unsigned integers or signed integers.
According to this Wikipedia page on 8-bit computing,
There are $2^8$ (256) different possible values for 8 bits. When unsigned, it has possible values ranging from 0 to 255; when signed, it has -128 to 127.
The natural answer, without no more context, should be $2^8-1=255$. We have not seen any computer processor that interprets an 8-bit sequence (in a basic level, just to be safer) as an integer that is more than 255. On the other hand, almost all computer processor today can interpret an 8-bit sequence as an (unsigned) integer as large as 255 . In fact, as said in this nice answer of Kristian H,
Unsigned numbers are one interpretation of a sequence of bits. It is also the simplest, and most used interpretation internally to the CPU because addresses, and op codes are simply bits. Memory / Stack addressing and arithmetic are the foundations of microprocessor, well, processing. Moving up the abstraction pyramid, another frequent interpretation of bits is as a character (ASCII, Unicode, EBCDIC).
Having said the above, let me come to defend the choice of the author of that exam, even though I would NOT phrase the question and answer suite as presented in the question.
The question in the exam was designed to test a student's basic understanding of the representation of numbers in our binary computer processors.
If a student chooses, presumably for some good reason, "b) (127)10", we can be confident that student knows the very basic about the representation of a signed integer by two's complement.
One bit out of that 8 bits must be used to represent the sign. Somehow, the range of positive integers is one less than the range of negative integers. So, the maximum signed integer in 8 bits is $2^{8-1}-1$. We can be confident that student would know the maximum unsigned integer can be represented in 8 bits could be $2^8-1=255$.
However, if a student chooses "d) (255)10", it might be hard to justify that student knows what happens with signed integers. So, if the exam is aimed at checking how much students have learned, choice b) could be appropriate.
So, in the context of an exam, a student would have a better chance to be considered more knowledgeable if choice b) is selected instead of choice d). Or a better chance to gain a better grade.
Once again, let me emphasize harold's verdict, "that is not a reasonable question". Ideally, for the intended choice of b), the question should be "the maximum signed integer that can be stored in memory of 8-bit word processor computer?"
For a detailed explanation of two's complement, please check this Wikipedia page.