"'Hello', says he. 'Hello', says I" — is this correct?
Solution 1:
Emilio,
"Says he... Says I" (pronounced sez hee/eye) was a very popular slang perhaps 40 - 50 years ago, and is most definitely not grammatically correct. Is is possible that the setting is that far in the past.
If the character is recounting a conversation he had in the recent past, proper grammar is, "said I" or "I said". If he is recounting wht he said in the present tense, "says I" is still incorrect. "Say I" is appropriate. But without more context, it's hard to say.
Sounds like a good book.
Solution 2:
English has many nonstandard dialects, and in some of them is is used as the present tense of be in all persons and numbers.