Answer: A lawyer is "effective" for this purpose as long as he or she is conscious for a "substantial portion" of trial. See, e.g., Muniz v. Smith, No. 09-2324 (6th Cir. July 29, 2011).
To prove ineffective assistance, a defendant usually has to prove that counsel's performance was deficient and that it actually prejudiced the defense. Strickland v. Washington, 466 U.S. 668 (1984). But some situations are considered outrageous enough that a court will assume there was prejudice. Is sleeping through trial one of those situations?
It is a little scary to consider that four federal circuits have now considered the question of "when sleeping by trial counsel becomes the effective denial of counsel" for this purpose. But they have, and the "substantial portion" test is the result. (A "not insubstantial" portion will also work, as the Fifth Circuit once ruled in a death-penalty case where defense counsel had been "repeatedly unconscious" during trial.)
The court in Muniz didn't have to decide exactly what "substantial" meant, because the only evidence Muniz had was an affidavit from one juror saying it was apparent that the attorney "was actually sleeping through a portion of his client's testimony." That portion was the portion where he was getting cross-examined, which seems like a fairly important portion, but the court held the standard wasn't met:
This [affidavit] alleges only that Muniz’s attorney was asleep for an undetermined portion of a single cross-examination. The record shows that Muniz’s attorney was not asleep for the entire cross since he objected near the end of the questioning. This is especially significant, given that the total cross-examination was fairly short, spanning only 26 pages of trial transcript. Muniz’s lawyer therefore must have only been asleep for a brief period. This is in contrast to [another case] in which the trial judge himself "testified that [defense counsel] 'slept every day of the trial.'"
He couldn't stay awake for the entire cross, no, but he did perk up there at the end.
So, this meant Muniz had to prove both parts of the Strickland test without the help of a presumption. The first part of the test asks whether the representation was deficient, meaning that counsel was not acting with "reasonable professional judgment." To its credit, the State did not try to argue that "Muniz's attorney fell asleep at trial because in his 'reasonable professional judgment' it was the best course of action." (Could unconsciousness ever be the best course of action? Depends. How badly are you losing?)
But Muniz still lost under the second part of the test, because he could not "show that there is a reasonable probability his counsel could have prevented either of [two allegedly] prejudicial events from occurring had he been awake."
I guess if things are going that poorly, you might as well try to get a little shut-eye.