In your example, the comma is used for clarity when the that is omitted.
Without the that and absent the comma, you effect a garden-path sentence, where (in this case) the noun following the linking verb wants to be a nominal subject complement before it gets around to being part of a zero that-clause.
For illustration purposes, I will use a different sentence as an example:
The odd thing is, this relationship may actually work.
Source: The Washington Post — Between Kid Quarterback and
Coach, Something Unexpected Has Come to
Pass
Let’s look at this three ways:
The odd thing is that this relationship may actually work.
The odd thing is this relationship may actually work.
The odd thing is, this relationship may actually work.
The second, garden-path version leads to The odd thing is this relationship. — leaving the reader to get back on the trail for the rest of the sentence.
Here are some examples from the Corpus of Contemporary American English. You can see that the comma-less, that-less uses are not very common.
the odd thing is
that
the odd thing is
,
the odd thing is
NOUN
the odd thing is
the
Circling back to your own example . . .
Believe it or not, the The odd thing is she. is a pretty good set up for a garden path, using — as it “properly” does — the nominative pronoun in the predicate.