I was thinking this was just a drop-catcher service, but I see now that it's much more than that:
Godaddy Domain Buy Service
Just $69.99 per domain name + commission*
Our Professional Domain Buy Service provides personal Domain Agents who will:
* Complete a Certified Appraisal within 2 business days of service purchase.
* Attempt to contact current domain owner, to determine interest in selling their domain.
* Negotiate the sales price of your desired domain (if the owner is willing to sell their domain)
* Help facilitate the sales transaction (if a sales agreement is reached).
*10% commission or $10.00 fee will be charged (whichever is greater) if the name is successfully purchased. Fee is non-refundable.
So what you are paying for is their time to see if they can help you buy the domain and if they can you have to pay them a 10% commission. Now a 10% broker fee is not bad, but in most cases the seller pays a broker to help them sell the domain. I don't know if this is as common, but it would make some sense, except for one thing: When a seller pays a broker, they pay them more commission the higher the price is. In your case, you also pay them MORE commission for a higher price. The point is that they have a conflict of interest to get you the lowest price and get the highest fee for themselves. That fairly sucks and is yet another reason I don't like Godaddy.

They should get a higher fee for the LOWEST price they can get for you...!
So for your money you are getting an appraisal to have some idea what the domain may be worth on the open market (Check Estibot.com first). Then you are getting them to contact the owner (if they can). They negotiate for you which may or may not be good depending on their ability (and motivation, see commission above). And they fasilitate the sale which I take to handle any escrow, but find out what that means first, there may be more fees to pay.
If you decide to use Godaddy, I would see if you can get them to take a fairly low set fee. Point out that they are not going to be working in your best interest with a percentage of the selling price. Once you have the appraisal you can discusss a fee, or name some fees that increase as the selling price goes down. It's worth a try!
So, whatever you think the domain is worth, you should contact the owner using the whois information, and make an initial offer that is more than $10. How much that is will be up to you, but you will want it to be enough to get the attention of the owner. If you offer $500 for a $3,500 domain, that may get their attention and they will see if they can get you up to $2-3,000.
One thing you can do is start fairly low, then increase your offer every few days until the reply. If they are not interested but respond because they are tired of hearing from you then you can start from there and see if a deal is possible.