What statistical method should i use in the following scenario?

1. I want to give users a list of ideas that they can score. 
2. They can optionally choose the score an idea (to say how good they
think it is). So, not everyone will score every idea.
3. People can rank it as very bad (-2), bad (-1), neutral (0), good
(1), very good (2).
4. Each rating would contribute to an overall score for the idea.

The challenge I have is deciding what to do with ideas ranked as
neutral. I can calculate a simple score by adding or subtracting the
good and bad ratings. I'm guessing the algorithm should include the total sample
size (number of users who voted).

Why not use all positive integers? (1,2,3,4,5) Then have your mean
negative response range be around 1-2 neutral responses around 3 and
positive responses 4-5.  It will be easier to adjust the way you read
your algorithm.

Suppose you have 2 ideas A and B. Suppose that user's votes (numbers
in parenthesis) are
idea A: "-2"(3), "-1"(2), "0"(6), "1"(4), "2"(2)
idea B: "-2"(0), "-1"(0), "0"(1), "1"(0), "2"(0)
Then the score (arithmetic mean) is equal 0 for both A and B. But
there is a slight difference between them. In the final list of
results you may show this difference by specifiing the numbers of
votes for each idea. It may look like
idea A: score = 0, number of votes = 17
idea B: score = 0, number of votes = 1

What I'm trying to get to is a simple numberic score that would be
simple for others to understand. Because I want people to rate
something as neutral I was wondering whether this approach would work:

1. Use positive and negative numbers to score the feedback.
2. Sum these up to get a score
3. Divide the final number by the number of people who voted.

So, for example, if 5 people vote on something:
3 vote positive (+1)
1 votes negative (-1)
1 votes neutral (0)

Total score = (3 - 1) / 5 = 0.4

Alternatively
4 vote positive (+1)
1 votes negative (-1)
0 votes neutral (0)

Total Score = (4 - 1) / 5 = 0.6

I wish I had paid more attention in my statistics classes at school.
Is this approach flawed?

Yes, it works. But you may want to emphasize that some neutral score
(for example 0.2) was obtained because of splitted discision and not
because people simply ignored that question. You may explicitly
specify the number of votes to make the difference betwen these two
possibilities obvious.

Thanks lophar-ga - I agree that some qualifier to show the total size
of the population that gave a vote would be helpful.

So, is that it? Anyone else have an alternative suggestions? If not,
does this approach have a name ?! Seems like it must be a cornerstone
of analyzing survey results.