Back in 1997, the WWE had a similar situation on its hands, where a one-dimensional face wrestler was receiving an overwhelmingly negative reaction to his clean-cut image.
That wrestler was the Rock, then known as Rocky Maivia.
Fans would chant "Rocky sucks" and "Die Rocky die," even though Maivia was supposed to be a good guy. Sound familiar?
Eventually, the Rock, an over-the-top, arrogant heel character, was born, but the character ended up having the reverse effect. Fans grew to love the Rock's catchphrases and promos, cheering him even as he would insult the audience.
This could work with Cena as well. If fans take to his new heel character, it could generate a positive reaction from the half of the audience that currently hates him.
If WWE turned him heel, Cena would no longer represent everything some fans see wrong with the present-day WWE, otherwise known as the "PG era." It would be a fresh start, a new chapter for Cena, and finally, something to cheer for.