Is a business card necessary for a graduate student?

As a PhD student, I had business cards and never used them. Now I don't even have cards. I consider them a relict from different times. I would say nobody strictly needs a business card anymore, and I rarely see them being exchanged at conferences either.

Now, a homepage is a different story. You definitely need a homepage, for many of the reasons you think you may need a business card.

It's probably discipline-specific, but I hand mine out all the time. That being said, I talk to lots of vendors of computer hardware that I want to get back in touch with, so exchanging cards is helpful. As a grad student, it might be worth having some small cards from someone like Moo that have your name, email, affiliation, and a link to your website. It might be easier to hand those out than to carry 20 preprints of your article when you're giving a talk.

All that being said, it'd be pretty unusual to push your cards on people you meet at conferences. If someone asks for your information, or a copy of your publication, you might offer the card, but I wouldn't volunteer it unless you've been asked for something. It's just not part of the culture in most of the fields I have experience with for students to do so.

If there's any chance that you're going to be interacting with industry at conferences or other events, you need business cards. It's part of that culture. If there's any chance you'll be interacting with anybody from the far East, you need business cards (and ideally, a briefing on the etiquette of giving and receiving them).

Within academia (and not in the far east), it may depend on your field. Coming from a commercial background I was surprised to find that they aren't used in the same "scattergun" way in mine; but it can still save time with scribbling details, and a good card, offered appropriately, may make meeting you stand out in the fog of somebody's post-conference memory.

Summary: it's worth having some, but try to judge what is "normal" before pressing them on people.