Algorithms for "mostly-SAT?"
This cannot be true, since you could use such an algorithm to solve SAT itself in polynomial time. Given a SAT instance $\varphi$, construct a much larger instance $\psi$ which contains $\varphi$ as well as many copies of the clause $x$, where $x$ is a new variable not appearing in $\varphi$. It is easy to find an assignment satisfying almost all clauses of $\psi$. Now run your algorithm. If it succeeds, then $\varphi$ is satisfiable; and moreover, if $\varphi$ is satisfiable then it should, indeed, succeed. Thus $\varphi$ is satisfiable iff your algorithm finds a satisfying assignment within the promised running time.

Perhaps the participant meant an assignment which is close to a satisfying assignment, as in Schöning's kSAT algorithm; see for example Eppstein's lecture notes.