If we consider only the sun and the moon to be the determining factors, then No, the weight, even though by a negligible amount, would not vary. But, there are a lot of determining factors other than the Sun, and we know that there is no gravitationally neutral point in the universe, as the distance would have to be infinite, for in inverse square law to produce zero gravity.
The gravity of the sun is about 0.00589 m/s^2 at the radius of the earth's orbit, or about 0.0006 of the earth's gravity, g = 9.81 m/s^2