There are a few well-known insects in Greco-Roman myth, such as the gadfly that plagued poor Io after she was transformed into a cow and the fleas and flies of Aesop's fables. But the only (more or less) mythological source I can think of in which something like a mosquito figures prominently is the poem "Culex," attributed to the great Roman poet Virgil.

The eponymous culex is a "tiny nurseling of the moisture," but probably a gnat rather than a mosquito. It wakes a shepherd by biting him on the eyelid, and so saves him from a poisonous snake. The ungrateful shepherd squashes his savior. That night, however, he is visited by the ghost of the gnat, which criticizes him at length. A small sample:

Alas! how gratitude has gone aside

From kindly service, when I from the very door

Of death did give thee back to those above!

O where are piety's rewards and where

Its honors? (331-5)

The dead bug continues in this vein for some time, describing the underworld and its denizens. At last it concludes, rather plaintively:

Away I'm going, never to return:

Do thou, rejoicing, tend the pastures green

And woods of fount and groves. (571-3)

The shepherd wakes; and the following day he builds a nice little memorial for the gnat:

...and with a plenteous mound

Of earth a tomb arose in circle shaped.

Around it placing stone of marble smooth,

He plants it, mindful of his constant care. (590-2)

This might not be the sort of myth you were looking for, in the sense that it wasn't devised to explain anything (the whole story was almost certainly invented by the poet to show off his virtuosity). But it is at least an interesting story involving an irritating insect...

The mosquito has a small, albeit important, role in the Popol Vuh, the sacred text of the Maya-K'iche people from Guatemala.

[Hunahpú and Xbalanque] went on from there, and came to a crossway of four roads. They knew very well which were the roads to Xibalba; the black road, the white road, the red road, and the green road. So, then, they sent an animal called Xan. It was to go to gather information which they wanted. "Sting them, one by one; first sting the one seated in the first place and then sting all of them, since this is the part you must play: to suck the blood of the men on the roads," they said to the mosquito.

"Very well," answered the mosquito. And immediately it flew on to the dark road and went directly toward the wooden men which were seated first and covered with ornaments. It stung the first, but this one said nothing; then it stung the next one, it stung the second, who was seated, but this one said nothing, either.

The mosquito then proceeds to sting each one of the Lords of Xibalba, thus making reveal their names. And then comes a twist:

In this way they told their names, as they all said them one to the other. So they made themselves known by telling their names, calling each chief, one by one. And in this manner each of those seated in his comer told his name.

Not a single one of the names was missed. All told their names when Hunahpú puffed out a hair of his leg, which was what had stung them. It was really not a mosquito

Taken from another, earlier excerpt, which is a bit more humorous:

"Good," she said and went. Then they began to eat, but they were not really hungry; it was only a trick. They saw then by means of their plate of chile how the rat went rapidly toward the ball which was suspended from the roof of the house. On seeing this in their chile-sauce, they sent to the river a certain xan, an animal called xan which is like a mosquito, to puncture the side of their grandmother's water jar, and although she tried to stop the water which ran out, she could not close the hole made in the jar.

"What is the matter with our grandmother? Our mouths are dry, with thirst, we are dying of thirst," they said to their mother and they sent her out. Immediately the rat went to cut [the cord which held] the ball and it fell from the garret of the house together with the ring and the gloves and the leather pads. The boys seized them and ran quickly to hide them on the road which led to the ball-court.

After this they went to the river to join their grandmother and their mother, who were busily trying to stop the hole in the water jar. And arriving With their blowgun, they said when they came to the river: "What are you doing? We got tired [of waiting] and we came," they said.

"Look at the hole in my jar which I cannot stop, said the grandmother. Instantly they stopped it, and together they returned, the two walking before their grandmother.

And in this way the ball was found.