"Making a federal case out of it" is figure of speech (and had the same meaning in 1997 when the movie Air Force One was released that it does today).
It means making a big deal out of everything and going all out to formally address things through cumbersome channels even when this may be disproportionate to the actual stakes of a dispute. It does not always literally refer to an actual or possible federal court case, or even a literal court case at all. Sometimes it just means making a big deal out of something (when that isn't necessary).
This has its origins in the time period when you had to have a large dollar amount in controversy between litigants to bring a lawsuit in federal court under either federal question or diversity litigation (the limit now only applies to diversity litigation when there is no federal law dispute at issue, so there are actually many simple, small dollar lawsuits in federal court these days). The exact amounts at particular times with inflation adjustments to 2023 dollars are set forth in the footnote below.
Thus, historically only large dollar disputes could be brought in federal court and the litigation of those very big dollar disputes was always cumbersome and intense. So, the term "a federal case" came to be metaphorically associated with any case being litigated as if there were huge economic stakes involved, even if it was actually in state court and could never have been brought in federal court.
Almost always, the term is used in the sense of "don't make a federal case out of it", since for many minor disagreements that all out big dollar civil dispute style of intense litigation and formal legalistic fighting is inappropriate.
Dictionary.com claims that the metaphorical sense of the phrase had its origins in the time period from 1950-1955.
Wikipedia asserts (in sources compiled at the same link) that the phrase was popularized by:
New York City-born comedian Jimmy Durante (1893-1980) who used “Why
the guy’s making a federal case out of it” on his radio show with Gary
Moore, The Durante-Moore Show, broadcast about 1944 and printed in a
book published in 1945. The phrase was picked up by other New York
writers (Walter Winchell, Evan Hunter, George Axelrod, Jerome Weidman)
and it appears likely that Durante coined the expression of “making a
federal case.”
The link above corroborates this conclusion with references from Google Books.
Footnote Re Inflation Adjusted Dollar Amounts
Congress first established the amount in controversy requirement when
it created diversity jurisdiction in the Judiciary Act of 1789,
pursuant to its powers under Article III of the U.S. Constitution, the
amount being $500. It was raised to $2,000 in 1887, to $3,000 in 1911,
to $10,000 in 1958, to $50,000 in 1988, and finally to the current
$75,000 in 1996. . . .
Congress did not create a consistent federal question jurisdiction,
which allows federal courts to hear any case alleging a violation of
the Constitution, laws, and treaties of the United States, until 1875,
when Congress created the statute which is now found at 28 U.S.C. §
1331: "The district courts shall have original jurisdiction of all
civil actions arising under the Constitution, laws, or treaties of the
United States." At that time, such cases had the same amount in
controversy requirement as the diversity cases. Congress eliminated
this requirement in actions against the United States in 1976 and in
all federal question cases in 1980.
(Source)
Adjusting for inflation and rounding to the nearest thousand dollars:
$75,000 in 1996 dollars is currently worth about $143,000.
$50,000 in 1988 dollars is currently worth about $127,000.
$10,000 in 1958 dollars is currently worth about $104,000.
$3,000 in 1911 dollars is currently worth about $95,000.
$2,000 in 1887 dollars is currently worth about $63,000.
$500 in 1789 dollars is currently worth about $17,000.