DO BOYS OR GIRLS HAVE FASTER HAND-EYE COORDINATION?

It depends.

Some studies suggest differences between fine and gross motor skills.
e.g. from http://www-personal.umich.edu/~kruger/ep10.html

"Men excel at certain types of mathematical problems, are generally
better at reading maps, solving mazes, and completing other
visual-spatial-quantitative tasks such as mental rotations and space
relations (Silverman & Eals, 1992). These skills would be useful in
navigating about the landscape. On average, men are better at gross
motor skills requiring speed and force, from running and jumping to
throwing objects (Fisher, 1992).

Women on the other hand, average better "fine" motor coordination,
manipulating tiny objects with ease (Fisher, 1992). They are also
usually superior at spatial locations and object memory tasks,
valuable while searching the underbrush for resources (Silverman &
Eals, 1992). Although there is more variation within the sexes than
between the sexes in tests of verbal abilities, on average girls;
speak sooner, more fluently, with greater grammatical accuracy, and
with more words per utterance than boys (Fisher, 1992).
references from the quote can be found at 
http://www-personal.umich.edu/~kruger/evref.html

The cause(s) of any reported differences is a very different question.

Girls seem to enjoy purely coordination games: Jacks, rope skipping,
hopscotch; whereas boys' games of coordination are more directly
competitive: marbles, etc. (for lack of immediate examples).  The
girls compete, too, of course, but on points, not on direct
elimination.
Do the girls' games suggest a greater aptitude and hence interest, or
a need to improve weak native skills through training?
Embroidery is a fine motor skill ascribed to girls in the Western
World, but in the Orient it is (has been) a male skill along with that
of weaving Cashmire shawls, and jewelry and watch making making in
both cultural areas.

None of this contradicts the previous comment.
I think it is more a question of nurture than nature.