‘Caret’ is the word you are looking for. I do believe though, that it is part of the browsers design, and not within the grasp of css.

However, here is an interesting write up on simulating a caret change using Javascript and CSS http://www.dynamicdrive.com/forums/showthread.php?t=17450 It seems a bit hacky to me, but probably the only way to accomplish the task. The main point of the article is:

We will have a plain textarea somewhere in the screen out of the view
of the viewer and when the user clicks on our “fake terminal” we will
focus into the textarea and when the user starts typing we will simply
append the data typed into the textarea to our “terminal” and that’s
that.

HERE is a demo in action

2018 update

There is a new css property caret-color which applies to the caret of an input or contenteditable area. The support is growing but not 100%, and this only affects color, not width or other types of appearance.

input{
  caret-color: rgb(0, 200, 0);
}

<input type="text"/>