Chances are that the swag would actually cost more than a free $6 game (Ex. t-shirts are usually $15-$20). Really, I think a coupon code would probably best option. I'd be happy with a simply email saying happy birthday like most forums send out automatically.

An e-card on your birthday would be nice.
It can even be nicer if it's based upon 1 or more of the games you bought.

If you bought Fallout, would it say "Yours Apocalyptically"? "Have a BLAST?"

Having just purchased Screamer, it would be all in caps?

For In Cold Blood, it could be a drugged up Cord reading the birthday message at gunpoint

I don't think it's very common (or a good idea) for businesses to give out free stuff to their customers on their birthdays (though I'll try getting some free food at my local grocery store next time), but I do actually think a birthday discount could potentially work to GoG's advantage. Discounts make people happy, and they generate sales.

Good customer relations I guess. Although, for it to be worth their while, specific criteria should be in place first. Such as only being able to enter your birthdate once and not be able to edit it later (a large number of sites do this already).
Then, people should have to be a member of GOG for at least one year and have at least X number of games purchased on their account (say, at least 10). And even then, any free game should only be at the lowest price scale.
A better idea for GOG would be to have a discount system in place instead. Where on your birthday, you'd have 10 to 50% off of one game. Same restrictions as before should be in place though. (Ie, member for one year minimum, at least X number of games in account). Different percentages off could depend either on years as member or on number of games already purchased. Rep, however, should not be a deciding factor in any way (avoids potential abuse).
Personally, I prefer the percentage off model myself too, as opposed to just handing out a free game.

But then, GOG's running a business that's really not making a lot of money - $2 says they're using funds off The Witcher. Personally I appreciate the fact that the games here are all fuss-free, and the fact that there are so many weekend promos on games that already cost $5.99.
I'd be quite guilty (though very happy) if GOG gave discounts on games simply because it's your birthday, but I think guilt overrides pleasure for me heh
By the way, hasn't it occurred to anyone that people could simply create new accounts to obtain "free games"?

Yeah thats a point, I like the minimum number of games idea as long as its at least 3 so people can't just have lure of the temptress & beneath a steel sky.
Hmm, my local music shop has a buy '10 get 1' free thing, that'd be an interesting thing to have for a site like GOG.

That's why I suggest a minium time frame and minimum number of purchased games. Obvoiusly, the 2 free games should not be included within that count, to avoid people just grabbing those in an attempt to get more for free. It's also why I think a discount would be better (but still with the same restrictions).

It still can be exploited by people who create multiple accounts and then wait a year. If I was a scammer and had the sole intention of milking GOG for all it's worth, I'd probably create a few accounts and then wait it out, knowing that I won't be able to get anything yet, but in a year's time, I can get 10 free/discounted games from the "fake" accounts that I make
I'm not being anal about the idea, but I'm just saying that FWIW, it's just not going to work out without people being assholes and exploiting the system - the same thing as what pirates are doing today

Yes, it can, but if you take the "minimum number of games in account" rule into consideration, who'd want to cheat? Would you really make extra accounts, and buy a number of games for each of them, just to save 3$ on a game? If you would, then go ahead. It'll still mean more sales for GOG :-D

True that makes sense, provided GOG still gets a slice from the cake ie. profit ;)

Oh, I like this idea much better than "a free game for your birthday". This one I could see happen.