So why do they do it? The group appears, based on its past attacks, to have pretty simple motivations: attention for itself and punishment for Western media organization they perceive as biased against Syrian President Bashar al-Assad. The Syrian Electronic Army actually makes a lot more sense if you think of them as pranksters who also happen to love Assad than as state-aligned hackers in pursuit of concrete goals.
The effect of the hacks is typically not to steal information or sabotage institutions, but rather to hijack the targeted outlet for a few minutes, plastering it with the group's message and perhaps some condemnation of U.S. policy toward Syria. Their hacking power seems to exist purely to demonstrate their hacking power, taking down popular sites purely to claim credit for it. To paraphrase the great Web comic XKCD, it's less like breaking into the New York Times than defacing the New York Times.
Past hacks have posted social media "memes" or other large-font messages, typically in colloquial English, lampooning Western media and fist-pumping on behalf of the Assad regime. It's yet another indication that the Syrian Electronic Army is far from some Damascus-run formal Syrian enterprise, but more likely an informal network of young volunteer hackers; slacktivists for a bad cause.
As my colleague Caitlin Dewey has written, hacks like this are good for publicity – the SEA even released a dramatically soundtracked video of one attack — but they don’t have much staying power because the institutions quickly see the hack and recover. The hacks get a lot of attention and that's pretty much it.