My cynical suspicion is that the lotus position is traditional, and looks cool, and therefore people have come up with rationalizations for it.
Someone has already quoted Wikipedia, so I won't repeat it, but most of those sound like benefits that you can get by lying down, or finding a really comfortable chair, or something similar. And the bit about improving digestion sounds especially dubious, since it doesn't really say what that means, and a lot of Yoga folks have a bad habit of making vaguely shiny-sounding medical claims without evidence.
(Do I sound negative? I actually do sometimes meditate, because it's a pleasant thing to do. I just want to make sure I'm not fooling myself about why I'm doing it.)
Speaking from my own experiences, and pulling loosely from the zen tradition in which I practice:
Sitting meditation keeps you more alert, makes it easier to reach a state of samadhi, or focused, non-discriminating awareness. For a while, due to back pain, I switched to lying meditation. I was not a fan, it was much easier to drift off. Zen meditation typically involves leaving the eyes partly open to avoid daydreaming or entering some sort of trance state. I also found this difficult to maintain while in a prone position.
Sitting in full lotus is difficult for me, and I'm relatively flexible. I generally sit half-lotus or in a chair. Sitting without a chair for long periods causes pain in the legs and back/shoulders. That's part of the practice, learning to maintain focus and equanimity through pain and discomfort.
To the best of my knowledge, the common sitting postures - lotus, half-lotus, burmese, seiza - came about because they were maintainable for long periods. Sitting indian style with legs crossed in front, for example, your stomach muscles will quickly tire and you'll be hunched over.
I did quote Wikipedia and even though I'm (definitely) sceptic about the digestion bit myself, the others do make at least some sense. Obviously it is always quite easy to find justifications for something you believe (or want to believe) in.
I personally don't believe meditation needs a specific position. I've seen and read a few things that document the practice while walking and while lying down. I guess tradition does point to the lotus being the normal way to do it, but as long as you get the benefits of it in whatever position you find yourself in, you're fine.
I apologize for simply lifting this off Wikipedia, but I too had that question before, and this paragraph was helpful:
The Lotus position is adopted to allow the body to be held completely steady for long periods of time. As the body is steadied the mind becomes calm, the first step towards meditation. The posture applies pressure to the lower spine which may facilitate relaxation. The breath can slow down, muscular tension decrease and blood pressure subside. The coccygeal and sacral nerves are toned as the normally large blood flow to the legs is redirected to the abdominal region. Digestion may also be improved.
My understanding is that the lotus position forms a tripod or your two knees and your butt, which is supposedly pretty stable.
I wouldn't know, as I haven't a prayer of attaining that position.
I have a little folding wooden meditation bench that I use. I don't think I'd want to meditate for more than half an hour using it, though. Some similar benches are available with cushions.
Someone has already quoted Wikipedia, so I won't repeat it, but most of those sound like benefits that you can get by lying down, or finding a really comfortable chair, or something similar. And the bit about improving digestion sounds especially dubious, since it doesn't really say what that means, and a lot of Yoga folks have a bad habit of making vaguely shiny-sounding medical claims without evidence.
(Do I sound negative? I actually do sometimes meditate, because it's a pleasant thing to do. I just want to make sure I'm not fooling myself about why I'm doing it.)