The swan is seen as an ideal of the self-realised person (the ideal for real yoga practice and its only raison d'etre for the practice ctually) because it is seen in Indian symbolism as the perfect detached creature -- it is in the water, yet its feathers are not we. It also can walk, fly and swim. The word ham-sa when transposed sounds so-ham -- this is the sound of your own breath, according to yoga. It is the sound that is manifest in the body, psychically experienced by it. It may explain why when people meditate on this sound (which is called the nirguna mantra, or the chant of the formless) they feel as if they have entered a trance state (one contemporary guru uses it as part of his practice and some silly followers actually think he made up the sound. One such follower once sent me a threatening letter when I wrote a yoga column on this sound and its meditation, which btw is called ajapa japa famously in all of yoga and does not belong to any one group, or its leader!!).
So, so-ham and ham-sa mean the realised being -- it is an experience of the ultimate one which even Patanjali says is why nature conjured us up, and gave `a vague nothing a local habitation and a name' (as Shakespeare talks, in Midsummer's Night Dream). So this sound is also the sound of your breath.
That it should be the vehicle of Saraswati means that wisdom rides on your breath (your awareness of it)-- more on this later:)
Interesting thing is that when u do the hamsasana (a preparatory pose to the peacock) your breath stops altogether!!