How strong of a magnet would be needed to mess up this hardware?

I just bought a magnetic dry erase board that also came with 2 small magnets. I intend to carry these items everyday in my backpack alongside my laptops and my phone. Some days, I will have my Lenovo Yoga 2 Pro in there, some days my Alienware M17x R3, and otherwise just my Galaxy S3 phone in a small pocket.

Do I have anything to fear in regards to damaging the hard drives on these devices? I'm not sure if the magnetic strength of a dry erase board is anything to fear...

You have nothing to fear. There is no magnetic storage in your phone, so that one is easy.

For hard drives, or any magnetic thing, the property that is important here is called coercivity, which is the resistance of something magnetic to changes in its magnetism. That is, how strong of a magnet do you need to overcome the coercivity of the hard drive?

For modern hard drives, even powerful neodymium magnets are not enough to demagnetize. The same is not true for low coercivity things like credit cards and magnetic tapes. You're much more likely to physically damage the drive by bending the heads by attracting them with a powerful magnet than you are to demagnetize the platters. And that's with rare earth magnets.

The weak magnets that come with white boards or that you stick on your refrigerator? About as worrisome as a cloudy day.