It appears to be a simple case of a hand-written J looking too similar
to an I:
"There are many legends about why. The most prevalent, according to
the Historical Society of Washington, D.C.: Even after J was added to
the alphabet, it looked too much like I, and those who laid out the
city thought it would cause great confusion among travelers."
Q: WHY IS THERE'S NO "J" STREET IN THE DISTRICT?
A: WE CONTACTED THE WASHINGTON HISTORICAL SOCIETY FOR YOUR ANSWER. IT
SAYS, FOR MUCH OF THE COLONIAL PERIOD THERE WAS NO LETTER "J" IN THE
(PRINTED VERSION OF THE) ALPHABET.
AFTER "J" WAS ADOPTED, THE LETTER REMAINED OFF DC STREETS BECAUSE "I"
AND "J" , WHEN PRINTED, APPEARED IDENTICAL AND WERE CONFUSING.
FOR MORE INFORMATION ON STREET NAMES AND OTHER INTERESTING DC FACTS
VISIT THE WASHINGTON HISTORICAL SOCIETY NEAR DUPONT CIRCLE -- OR CALL
IT AT (202) 785-2068.
BY THE WAY, "J" IS NOT THE ONLY LETTER MISSING FROM D-C STREET NAMES.
WHEN L' ENFANT PLANNED THE CITY, "W" WAS THE LAST LETTER OF THE
ALPHABET. AS A RESULT DC ALSO LACKS X, Y AND Z STREETS.
"The most plausible explanation is that J Street was omitted because
the letters I and J were often indistinguishable from each other
(especially when handwritten), and in 18th century English they were
still largely interchangeable. (The 1740 "New General English
Dictionary" published in London had a single section for I and J, and
the standard identification Thomas Jefferson used on his personal
possessions was "T.I.") Having both an "I" and a "J" street would have
been redundant at best and confusing at worst, so "J" ended up as the
odd man out. "
More on your question is on this web page:
Urban Legends Reference Pages: History (No Way, No Jay)
letters alphabet Washington DC streets