Perhaps it would sound more familiar and make more sense if you rearranged the words to say: Shall I tell you something I remember? "I'll tell you, shall I, something I remember?" sounds literary, like something you'd find in a book, but would not be common phrasing in everyday life.

The sentence is a bit redundant sounding. " I'll tell you, shall I, something THAT I remember" sounds better to me. The THAT seems to make the sentence flow better. I think "shall I" is the redundancy and i would just leave it out of the sentence. It adds nothing to the overall meaning anyway.