My answer is based on oral transmission from Vajrayana teachers and thus I am unable to provide references.
I've been told that every meditation, however short, will bring about positive results and the only worthless meditation is the one you never did. Imagine that over a period of a month one could sit only for one minute a day but never did thinking that it is worthless. If they actually sat down, they would have done 30 minutes of meditation in total, which, admittedly, is nothing particular, but they would also form a great habit by doing one thing 30 days in a row. It means that 30 times they stopped doing their worldly activities and sat down to change something in their life.
Keeping a meditation session short is actually a common advice for beginners. Meditation should not be perceived as a difficult duty but rather as the best gift one can give to oneself. If one starts off too ambitiously, after a week they would connect 'meditation' with 'sore back', 'confusion', 'uncontrollable thoughts', 'drowsiness' etc. To prevent this, it's better to sit for a few minutes and finish the session while still feeling fresh. Then one will form positive connotations about meditation and will willingly sit down next time and possibly won't be bothered too much with prolonging the sessions step by step.
In Buddhism, one is not interested in immediate results as following Buddha's teachings is a long journey and many lifetimes might pass before one reaches enlightenment. One should do whatever is possible in the current situation and everything that is done with the motivation to follow Buddha's path should be considered productive.
Practising meditation is a very individual issue and coming up with a universal minimum value wouldn't help anyone. Everyone meditates as much as they can and want. If one can only do one minute a day due to a busy schedule, all they can do is to possibly make wishes that soon their conditions will change and they will be able to find more time. There are cases, however, that the reason why people meditate only a little lies elsewhere. They are again busy, but this time with things like browsing Internet, meaningless chatting, gossiping, napping etc. My point is that there is no use making a hard rule about meditation time, rather one might reflect on why they meditate that amount of time and not any more. Checking one's motivation and contemplating on impermanence might help in deciding whether one wants to meditate more.