How should one manage the Bitcoin blockchain on a Linux laptop with limited storage? I want to run Bitcoin-Qt on a laptop that has limited storage (it has an SSD and is used for many purposes that all require a bit of hard drive space). Assuming these hardware limitations, what would be a good way to manage the blockchain?

So, to be clear, I cannot store the blockchain on the laptop and I want to be able to make regular transactions portably on the laptop.
Starting with Bitcoin Core 0.12.0, due out in Jan/Feb 2016, you'll be able to add the prune option to your configuration file to only store the most recent blocks.  For example, if you specify prune=5000, you'll only store as many of the recent blocks as fit into 5,000 MiB of disk space.

On top of that, you'll also have to store the UTXO database, which is currently about 1.2 GiB and growing, plus maybe a few megabytes of other things like the binaries themselves and your wallet.  On Bitcoin.org, we're saying that 5 GiB is a reasonable minimum for now.

Note that Bitcoin Core 0.11.* includes a prune feature, but in those versions it disables the wallet.  (This feature was initially targeted at miners who use mining software to manage payout addresses, and it was considered unwise to have regular users use the feature until it got more testing.)

If I were you, I would consider the feature somewhat experimental and make sure to maintain backups of the wallet (which is good policy anyway, of course).