Is there any scientific evidence that very short (e.g., 1 minute) meditation sessions have positive outcomes (e.g., improved focus)?

The mechanism is like this: Wherever you place your attention to, your energy flows. So, if you sit and genuinely meditate for 1 minutes, for 1 minute you will direct your energy towards your awakening. But, if the rest of the day you revert to letting your focus go and play on its own, your energy will continue empowering your subconscious mind again, which is the autopilot serving until you take conscious control of the human system. But even that 1 minute will make a difference, if for that one minute you truly commit to practicing by the instructions. It will start changing your understanding. You will begin to see the difference between living on autopilot, and living meditatively. But make it count. If by your free will you are choosing to meditate for even 1 minute, give your best in that 1 minute.

This can be tricky to study scientifically because it is similar to building a lie detector. A person can deceive themselves that they are meditating in that one minute and tell that to the scientists. Or a person may be genuinely meditating and tell that to the scientists. All that the scientists can do is to measure the metrics related to the body. There are devices even able to measure the emotional system, strength of the energy centers (chakras). These can be tracked by scientists to understand if some change is happening. I heard from my teacher that they measured his chakras in India.

You can make your own science experiment yourself. Take a journal and start recording your expressions about your state after sincerely meditating for 1 minute. Track your progress day by day. If you truly are willing to look, you will notice the changes. Sincere meditation itself will make changes that are quite impossible to ignore as long as you keep going.

Even short, truly sincere meditations can give very deep insights. But ultimately, we are not trying to escape from this world with a 1 minute meditation. You can even explore/focus into your deepest aspect in one minute, and it will be a tremendous insight but you must learn also live from it while living in this world, which is meditativeness. Simply focusing on the object of your meditation sincerely will result in progress.

In conclusion, any sincere effort, any focusing in that direction will keep causing a transformation towards awakening, even though at times the transformation being undergone might not be immediately noticable to the practitioner.