What does sati mean? This normally gets translated as mindfulness.

In particular, does it mean remembrance, as in to actively think using words about the past. 

Or is it about thinking about the present, and not the past.

Or is it about suppressing ones internal dialog.

If the meaning has changed, is it a relatively modern thing, or does it trace back to some sort of new development in the history of Buddhism.
The word Sati (Smrti in Sanskrit) literally means remembrance or recollection, but as with many Buddhist technical terms has a more specialized meaning in a meditative context. In the context of meditation the word Sati has a more general meaning to set the mind on the object of meditation. For example, the meditation in which one contemplates the attributes of the Buddha is called Buddhanusati.

However quite often when people say Sati, they mean a specific kind of Sati. That is to say, Samma-Sati, or right mindfulness, which is the seventh factor of the Noble Eightfold Path. The Suttas define Samma-Sati as follows:


  And what is right mindfulness? There is the case where a monk remains
  focused on the body in &amp; of itself — ardent, alert, &amp; mindful —
  putting aside greed &amp; distress with reference to the world. He remains
  focused on feelings in &amp; of themselves... the mind in &amp; of itself...
  mental qualities in &amp; of themselves — ardent, alert, &amp; mindful —
  putting aside greed &amp; distress with reference to the world. This is
  called right mindfulness.


(http://www.accesstoinsight.org/tipitaka/mn/mn.141.than.html)

So with right mindfulness one is specifically directing the mind to the body, feeling, mind, and mental qualities. The word which the translator translates as Alert is the term Sampajanna, which others often translate as "fully aware" and quite often when people say mindfulness, they are actually thinking of Sampajanna although they aren't totally distinct, as you need Sampajanna to have Samma-Sati.