Is there an option to change the space/distance between tick labels and axes labels for plots?

The most convenient way I found is to wrap Plot (without FrameLabels and PlotLabel and with appropriate ImagePadding and ImageMargins) inside Labeled and use the Spacings option to position the labels:

 Labeled[Plot[Sin[x], {x, -Pi, Pi}, Frame -> True, 
 ImagePadding -> {{20, 1}, {15, 2}}, ImageMargins -> 0, 
 FrameTicksStyle -> 10, RotateLabel -> False], 
 TraditionalForm /@ {Sin[x], x, Row[{"This is a plot of ", Sin[x]}]}, 
 {Left, Bottom, Top}, 
 Spacings -> {0, 0}, LabelStyle -> "Panel"]

This can also be achieved by

• encasing the graphic inside a Show,
• setting the outer Show's PlotRangeClipping to False, and
• adding the labels as Text commands inside an Epilog.

Thus, for your example, you would do

Framed[
 Show[
  Plot[Sin[x], {x, -Pi, Pi}
   , Frame -> True
   , FrameLabel -> {{None, None}, {None, 
      Row[{"This is a plot of ", Sin[x]}]}}
   , ImagePadding -> {{55, 10}, {35, 20}}
   , ImageMargins -> 0
   , FrameTicksStyle -> 10
   , RotateLabel -> False
   ]
  , PlotRangeClipping -> False
  , Epilog -> {
    Text[x, {0, -1.4}],
    Text[Sin[x], {-4.12, 0}]
    }
  ]
 , FrameMargins -> 0
 ]

which produces

You can then freely place the label text anywhere within the image box by adjusting the Text coordinates.

This has the advantage that (as far as I can tell, on v10.1.0) the resulting styling is identical to that produced by setting the labels via AxesLabel or FrameLabel; if you want to style them directly then you can do that as well.

Further, this produces an actual Graphics object, as opposed to an object with a Labeled head, which can be advantageous for manipulating and exporting the resulting plot.

Due credit to David Park's answer on this comp.soft-sys.math.mathematica thread for pointing out the technique.

When using SciDraw, we have full control over all label positions. The key option names are variation on TextNudge. The value specified in this option will be added to the label position.

Example with frame labels:

Figure[
 {
   FigurePanel[{}, FrameLabel -> {x, y}, 
     XFrameTextNudge -> 10,     (* shift up by 10 pt *)
     YFrameTextNudge -> {10, 0} (* shift right by 10 pt *)
   ]
 }
]

SciDraw of course draws its own frames and requires working in (and learning) its own framework. It will not help in changing the position of standard frame labels.

The ability of SciDraw to move labels is one of the several reasons why I usually use it for publication figures. I you have the time and are willing to learn it, it is a good option to deal with the label positioning problem.