It's not "I'm your Hucklebearer". It's "I'm your huckleberry."
"Huckleberry" was commonly used in the 1800's in conjunction with "persimmon" as a small unit of measure. As a result, "huckleberry" came to denote idiomatically two things. First, it denoted a small unit of measure, a "tad," or a person who was a huckleberry could be a small, unimportant person, usually expressed ironically in mock self-depreciation. The second and more common usage came to mean, in the words of the "Dictionary of American Slang: Second Supplemented Edition" (Crowell, 1975).
"A man; specifically, the exact kind of man needed for a particular purpose. 1936: "Well, I'm your huckleberry, Mr. Jones." Since 1880, archaic.
The "Historical Dictionary of American Slang" which is a multi volume work, has about a third of a column of citations documenting this meaning all through the latter 19th century.
So "I'm your huckleberry" means "I'm just the man you're looking for."