This question was actually really popular when Episode II came out, but I'm bringing it out now because I still hear it in whispers here and there.

When Uncle Owen visits the sandcrawler in ANH, why doesn't he recognize C-3PO? (And, presumably, R2-D2... though I don't hear that asked.)

That's a natural question... after all, C-3PO lived on the Lars farm. From the films, we don't know exactly how long he lived there, but it could have been as long as ten years.

Appearance
Physical appearance is the most obvious method by which most beings are recognized. This is most likely the main reason why Owen wouldn't know him.

We can tell from other protocol droids depicted in the films (TC-14, K-3PO, and the rude droid on Bespin, for example) that C-3PO's exterior appearance was extremely standard. He has no distinguishing physical features that would give Owen a clue. Color seems to be the only difference.

When Anakin took C-3PO from the homestead (without asking, I might add), the droid had a hodge-podge collection of beat up grey / silver coverings. Owen never saw 3PO with gold coverings.

An interesting point, though not really admissible in an exegetical discussion, is that in the original script for Episode II C-3PO was still in his "naked" Episode I appearance when Padme and Anakin arrive. Padme later found a box of coverings and was the one who put them on while waiting for Anakin to return. While those scenes were filmed, the concept was later discarded.

The deleted scene does raise another valid point to our discussion, though. It is possible that C-3PO spent some, or even most, of his time at the Lars homestead in his naked form. First impressions can be the most lasting, and perhaps that is the form Owen best remembers of his step-mother's protocol droid.

Notice that Owen does immediately recognize what C-3PO is. "You, I suppose you're programmed for etiquette and protocol." This language along with the dismissive tone indicated a lack of respect, at some level, toward the entire class of protocol droids. It is unlikely that someone with such a dismissive attitude toward a group would look more than superficially at an individual member of said group.

Name
So what about when 3PO introduces himself, "I am C-3PO, human cyborg relations..."? How many C-3POs would Owen have encountered?

BUZZ... trick question. Despite the protocol droid's usual routine of getting his name in immediately, he is never actually introduced by name to Owen in ANH.

OWEN: You, I suppose you're programmed for etiquette and protocol.
THREEPIO: Protocol? Why, it's my primary function, sir! I am well-versed in...
OWEN: I have no need for a protocol droid.

This is why C-3PO has to introduce himself in the garage to Luke, but Owen is not around. Later in the blue milk dinner scene, Luke never mentions the names of the droids to Owen. After Luke leaves the table, Owen never sees the droids again.

Voice
Each protocol droid that speaks in the movies has a different voice. It is possible that every droid has a unique sound and that Owen should recognize it. However, the practicalities of mass manufacturing would seem to dictate a finite selection of voices that could be selected for a given model. There isn't enough evidence to be conclusive on this matter.

Shared Background
If one learns that they have a shared past experience with someone (say, common town, high school, hobby or club), that can trigger the brain to make other connections and rediscover or recognize the other person by putting the puzzle pieces together.

Owen seemingly randomly blurts, "What I really need is a droid that understands the binary language of moisture vaporators." Does Owen recall that his family once had a protocol droid who programmed moisture vaporators? If not, this might be quite an unreasonable query for a class of machine seemingly not made for such labor.

Threepio replies, "Vaporators! Sir - My first job was programming binary load lifters... very similar to your vaporators in most respects." Since the droid's memory is wiped at the end of Episode III, this first (from lobotomized 3PO's point-of-view) binary load lifter job likely took place on the Tantive IV.

Even if Owen were to have a suspicion that this droid is the same one that his step-mother brought to the farm, Owen would know that 3PO's REAL first job had nothing to do with load lifters (would that first job be shining podracers?) and the 3PO he knew would know the binary language of moisture vaporators.

Since Owen would have no reason to think the droid to be lying, this piece of contradiction would have severed any connections being formed in Owen's head.

Time
Of course, there's the obvious argument that it's been about 20 years since they last saw each other, so given Owen a break. Maybe he's just a forgetful guy.

Conclusion
Different color, generic face, no name, conflicting backstory... I say this is very reasonable behavior. Not a plot hole.

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One parting thought, if I may. When Luke says, "He says he belongs to someone called Obi-Wan Kenobi," Owen's reaction is clearly one of suppressed emotion and, I think, putting some pieces together in his mind.

Is it possible that in that second, Owen's mind flashes backward in time to the very first time he ever hears the name Obi-Wan Kenobi... during the funeral of Shmi Skywalker from the voice-chip of C-3PO with R2-D2 at his side? "It seems that he is carrying a message from an Obi-Wan Kenobi." Does that memory start the farmer connecting the dots?

When Owen blurts out, "That wizard is just a crazy old man," is he actually thinking, "Oh man, these droids could expose the lies I've been telling this kid all his life"? That's the fear that motivates his orders, "Tomorrow I want you to take that R2 unit into Anchorhead and have its memory erased. That'll be the end of it." Or, "That'll get rid of the evidence."

The scene plays a little more interesting if you think of that connection happening and that subtext beneath the dialog.

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Continued in: SW Exegesis: How can Leia Remember Padme?