Is there any significance in Capt. Jack Sparrow being called Mr. Smith in the first Pirates of the Carribean?

I'm not sure if it has some special significance in this particular instance, but "Smith" is one of the most common surnames in the US, perhaps even the most common. So when someone wants to pick a pseudonym to mask their true identity, it's a typical choice. And for some reason, "Smith" has acquired this reputation more than any other common surname, so you'll frequently see it used for that purpose in TV and movies.

Adding something here,

When Jack first arrives at Port Royale, following conversation happens with harbormaster and Jack.

Harbormaster: Hold up there, you. It's a shilling to tie up your boat at the dock... and I shall need to know your name.

Jack Sparrow: What do you say to three shillings and we forget the name.

Harbormaster: Welcome to Port Royal, Mr. Smith.

Being Smith is very common and widely used name at that time, similar concept used just like hide something in a plain sight.

So Jack accepts that and he kept using name Smith to hide himself from others while appearing publicly.

Update..

In Pirates of the Caribbean : on Stranger Tides Jack impersonates a judge and he is also called as "Honorable judge Smith", This couldn't be a mere co-incidence.

As others have answered names like Smith and Jones are so common that they are often used as an alias, which is case with Captain Jack Sparrow docking on Port Royal...

However, the name Smith tends to be a thematic reoccurring name throughout the entire work...

1. Smith, like many surnames was once an occupational surname referring to either a lock smith or a black smith. So there's a bit of irony that Jack's fate would become entangled with the son of a former comrade, whose profession is a black smith. (Will Turner II)

2. In young Jack Sparrow adventure novels, Jack Sparrow is friends with a girl of similar age named Arabella Smith. There are also her parents, a Pirate named Laura Smith, and Laura's husband was a tavern owner of The Faithful Bride...

3. In Pirates of the Caribbean On Stranger Tides, Jack impersonates the real Justice Smith, whose been tied up and gagged.

4. In the fifth film 'Dead Man Tell No Tales', we are introduced to Carina Smyth, with her surname being a variation of Smith. In addition she is an astronomer who uses a gemstone, stars, under a blood moon to navigate and this juxtaposes a magical object known as the Sun-and-Stars Amulet that Arabella uses in the young Jack Sparrow Novels. The Trident also derives from the young Jack Sparrow Novels.

The harbormaster is indicating that he has accepted the bribe and will allow Jack to remain anonymous. He may also be slyly suggesting that he thinks Jack is up to no good. Giving the name of Smith when registering at a public accommodation is strongly associated in the public imagination with devious behavior.

For example, it is the name which adulterers supposedly write in hotel registers. Groucho Marx got a lot of mileage out of this in A Night in Casablanca. His character is working at the reception desk of a hotel when a Mr. and Mrs. Smith try to register. He loudly expresses the belief that "with a name like Smith" they can't possibly really be married.