So while there seems to be an increasing awareness of the value of attention, it is often only on the side of advertisers, selling our attention to companies, in order to boost these companies revenues. And it is working quite well. Obviously the advertising industry, lead by companies like Google and Apple, is in a bubble right now, but even before the over valuation of such companies advertising was a huge and for everyone involved profitable industry.
Actually there are some people like Cal Newport promoting the value of attention and uninterrupted focus. On his blog he has many examples of revered thinkers, who all made sure that they had uninterrupted blocks of time to pay attention to the problems they were solving.
But it is not only in business and science that attention is more valuable than time. Say you want to spend time with your favorite humans. If time was all that would count, then sitting in front of the television would make quite a pleasurable experience. However, obviously you just feel the same emptiness you did before, compounded with hate and disgust for the insulting pettiness and purposeful misinformation. No, you have to actually pay attention to the other person. Not just look in direction of their faces, but actually look at their face. Shut up the voices in your head and listen to what they are saying, instead of just smiling and nodding along.
Getting back to business, this also applies. Few things are less professional than to order people to meetings or worse, call them with the telephone machine and then not listening to them. Not only are you wasting their time, maybe even your worthless time, but also displaying the utmost disrespect. On the other hand, often you can help people simply by listening to their problems. Actively listening and asking questions, if something seems unclear.
I guess it is pretty obvious that attention is an important resource. So how can one quantify and measure it? Time can be easily measured using arbitrary fancy technology. Attention on the other hand, not so, at least on a practical level you have to resort to introspection.