What do you do to keep spirits high on your software team during moments of high stress or unpredictability? What is your morale-boosting secret weapon?

You have to start far before those moments.
Build resilience in yourself and your team, so that when the hard times come, there's a reserve to reach into.

0) You can't always be in "go-mode," so if high stress is the norm, something is wrong.

1) Ensure you're keeping up with your team, build trust, and learn who they are - what makes them tick. Not just to leverage it, but really learn to know your team.  

2) Communicate often.  Establish that you're going to keep your team up-to-date so there are few surprises (good or bad).

If you build that authentic trust, when you see a stressful time ahead, you can prepare your team.  Explain what's coming, whats needed and that you'll also be there with them.

First, I'd like to call out the importance of good planning, preparation, and setting expectations with stakeholders. If these are done well and consistently over time, the likelihood of being in an emergency situation decreases greatly.

When high stress situations do arise, these are the things I would suggest:
-Help the team understand "why" what they're doing is important to the organization, customers, etc. If they're bought in to the purpose, they'll find meaning in the work.
-Help them understand the timeline so there's an end in sight.
-Offer comp time off to help compensate for extra hours, weekends, etc.
-Give genuine praise for wins and milestones.
-As a manager, help and be present as much as you can.
-Consider rewards and recognition afterward, such as a small bonus, hand-written card, etc.