As described in How to define a figure size so that it consumes the rest of a page?, first define a measure to measure the remaining space on the page
\definemeasure[page][\dimexpr\pagegoal-\pagetotal-\lineheight\relax]
Then use a frame with its height equal to this measure
\framed[height=\measure{page}, width=\textwidth]{}
Now, to add a grid, you can create a Metapost background using
\startuseMPgraphic{page:grid}
....
\stopuseMPgraphic
and use it as a background for the frame
\defineoverlay[page:grid][\useMPgraphic{page:grid}]
\framed[background=page:grid]{}
Combing all this, we have
\definemeasure[topoffset][\bigskipamount] % We add this space before the answer
\definemeasure[page][\dimexpr\pagegoal-\pagetotal-\measure{topoffset}-5pt\relax]
% The 5pt is to avoid roundoff effects that can force the frame to the next
% page
% Add an option for grid size
\setupframed
[gridsize=10pt]
\startuseMPgraphic{page:grid}
begingroup;
newnumeric grid_size;
grid_size = \frameddimension{gridsize};
newnumeric x_count, y_count;
x_count := floor(OverlayWidth/grid_size);
y_count := floor(OverlayHeight/grid_size);
% The requested width and height may not a multiple of grid_size
% So, we center the grid horizontally and top align it vertically
newnumeric y_offset; y_offset := OverlayHeight - y_count*grid_size;
newnumeric x_offset; x_offset := (OverlayWidth - x_count*grid_size)/2 ;
newpath x_axis, y_axis;
x_axis := (x_offset, 0) -- (x_offset + x_count * grid_size, 0);
y_axis := (0, y_offset) -- (0, OverlayHeight);
pickup pensquare scaled OverlayLineWidth;
for i = 0 upto x_count :
draw y_axis shifted (x_offset + grid_size*i,0) ;
endfor ;
for i = 0 upto y_count :
draw x_axis shifted (0,y_offset + grid_size*i) ;
endfor ;
setbounds currentpicture to OverlayBox;
endgroup;
\stopuseMPgraphic
\defineoverlay[page:grid][\useMPgraphic{page:grid}]
% Define a new framed for answers
\defineframed
[answerframed]
[width=\textwidth, height=\measure{page},
frame=off, background={page:grid}]
\define\answer
{\blank[\the\dimexpr\measure{topoffset}]%
\answerframed{}}
\setupwhitespace[big]
\showframe % to visualize the result
\starttext
\input knuth
\answer
\stoptext
which gives

You can adjust the size of the grid using the gridsize parameter. If you want, you can also define a gridcolor parameter, and draw the rules using \MPcolor{\framedparameter{gridcolor}}.
[tikz-pgf]and "rest of page" etc. – Martin Scharrer May 27 '12 at 14:17