Pool their money into my own brokerage account and simply split the gains/losses proportional to the amount of money that we've each
contributed to the account. I'm wary of this approach due to the tax
implications and perhaps other legal issues so I'd appreciate
community insight here.
You're right to be wary. You might run into gift tax issues, as well as income tax liability and appropriation of earnings. Not a good idea at all. Don't do this.
Have them set up their own brokerage account and have them give me the
login credentials and I manage the investments for them. This is
obviously the best approach from a tracking and tax perspective, but
harder for me to manage; to be honest I'm already spending more time
than I want to managing my own investments, so option 1 really appeals
to me if the drawbacks aren't prohibitive.
That would also require you to be a licensed financial adviser, at least to the best of my understanding. Otherwise there's a lot of issues with potential liability (if you make investments that lose money - you might be required to repay the losses).
You should do this only with a proper legal and tax advice - from an attorney and/or CPA/EA licensed in your state. There are proper ways to do this (limited partnership or LLC, for example), but you have to cover your ass-ets with proper operating agreements in place that have to be reviewed by legal counsel of each of the members/partners,