Thank you for asking a very profound question!!
If you think about it the answer about whether the moon influences the apparent force of gravity has to be yes because the ocean has tides of about 0.5 meters that follow the moon. Answering the question of how much gravity is affected is much harder. The first question is how do you measure how strong gravity is? Then, how accurately can you measure gravity? And finally, is gravity constant over time? It was hard to find those answers, but the best I could do is:
accepted value for gravity is: ABOUT 981,000,000 microgals
approximate variation due to sun and moon is between 30 and 300 microgals, two thirds due to moon, on third due to sun
current instruments measure gravity changes to about 1 micro gal
gal stands for galileo; 1 galileo = 1 cm/sec^2; 1,000,000 microgal = 1 gal
Here is a paper describing how one group of scientists measured the force of gravity recently. It contains a figure showing gravity measurements over 18 hours, showing a variation of about 100 microgal due to sun and moon influences.
Here are 2 websites with lots of interesting information on measuring gravity, and how you can use that information:http://www.ngs.noaa.gov/GRDhttp://www.ngs.noaa.gov/GEOID
It is also very interesting to research the various ways people measure gravity.
This sounds like a really fantastic topic for a science fair project, though it may be most suitable for an advanced high school student.
Again, thank you for asking a really, really good question. I had great fun researching the answer over this weekend.