The helmsman could be weird...

Neither constructs, elementals, oozes, non-native outsiders, plants, nor undead need to sleep, therefore the helmsman could be a creature with any of these types or take feats or prestige classes that transform him into any of these types. However, a helmsman could also take...

• The general feat Tomb-tainted Vitality (Libris Mortis 31), among other effects, grants the creature the ability to go without food and sleep (yet the creature must still hydrate—whatever). The feat's prerequisites include the feat Tomb-tainted Soul (LM 31) and a nongood alignment. The feat gives the creature a "freakish skeletal appearance"—which seems a little judgy. Also, giving the helmsman these feats likely makes naming the helmsman Charon no longer optional.

...Or the vessel could be trapped (but in a good way)

Using the feat Craft Wondrous Item (PH 92) the vessel could have installed near the wheel a magical trap (DMG 74) that's a boon trap (Dungeonscape 135). Boon traps are horrible, notoriously both vague and broken, but this boon trap will be used only for...

• The 4th-level Clr spell remove fatigue [trans] (Book of Exalted Deeds 105) grants 1 touched creature/2 levels the benefits of 8 hours of restful sleep (but casters, if they need rest, must still get it). The spell removes the condition fatigued if present and permits the target to rest 1 hour to remove the condition fatigued if, before the spell, the target had the condition exhausted.

The DM must decide on the boon trap's precise effects, but it probably won't cost more than 50,000 gp, which, while expensive for such a trivial effect, really is a trivial effect. Taking the wheel'll trigger the boon trap which will cast the spell remove fatigue on the creature, letting almost any creature serve as overnight helmsman.

Trivia

There're no penalties for not sleeping unless sleep's necessary for something else, like preparing spells afterward. However, everyone knows sleep's a thing, so folks sleep even if the game doesn't mandate they must.

For those who will not be constrained by petty, demanding reality, Elder Evils, under the additional sign of the apocalypse Appaling Fecundity, presents some rules for sleep deprivation, saying that

A living creature can go without sleep for a number of days equal to its Constitution modifier (minimum one). Thereafter it is fatigued, remaining in this state for a number of days equal to its Constitution modifier (again, minimum one); if it would become fatigued during that time, it is exhausted instead. Each day after that period, the creature takes 1 point of Wisdom damage. If the total Wisdom damage exceeds its Hit Dice, the creature is affected as if by an insanity spell. (ElE 9)

The book then goes on to mention that during the Appalling Fecundity sign, one can't recover the Wisdom damage caused by lack of sleep (which is pretty clearly not the case for normal sleep loss). To clarify, the Wisdom damage accumulates until the creature falls unconscious, and lack-of-sleep effects (fatigue, exhaustion, Wisdom damage, insanity) are fixed only by sleep.

Although presented in the context of a Sign of the Apocalypse, these are decent (if long-winded) rules, finally published near the end of the game's run, for those who insist their characters needn't sleep. (Although anyone who insists his character needn't sleep will probably also insist these rules be ignored unless that sign of the apocalypse is present.)

Does a Construct need to sleep?
In the general case, no.

Does a homunculus need to sleep?
Probably not, but it's open to a DM ruling. It would be reasonable to rule that a Homunculus need not sleep, since it is a construct.

From MM pages 19 / 166: Constructed Nature (Animated Object / Golem)

"An animated Object (or Golem) doesn't require air, food, drink, or sleep."

MM page 188:

Homunculus, tiny Construct

A Homunculus is a construct."

MM page 6:

"Constructs are made, not born."

When we look at other constructs like Animated Objects and Golems, they don't sleep, so it's reasonable to rule that a Homunculus need not sleep and thus be a great alarm clock/alarm for a gnome who needs his beauty sleep.

Caveat: Modrons, Animated Objects, and Golems are not presented as having immunity to sleep, nor are homunculi. A save versus sleep might be necessary were the spell cast upon them. ("Sleep" / "Asleep" are not conditions. The sleep spell creates the Unconscious condition).

Counter Argument: since the line "constructed nature" is not included in the description, as it is for other constructs, it isn't treated as a construct for that particular case of "constructed nature."

Recommended Ruling: it does not need sleep since it is a construct.

Put another way: you can put a Homunculus to sleep, but it doesn't need sleep. (No amount of beauty rest will make it any prettier. It's an ugly baby that only its creator could love ...)