There are all sorts of programs available for "eBay sniping" - automatically placing your bid in the last moments of the auction in order to minimize the amount of information available to your adversaries (the other bidders) and thereby maximize your chances of winning while minimizing the amonunt you expect to pay.
The trouble with this is that it creates two classes of eBay bidders - those who have paid extra (in money, effort or both) for the sniping software, and those who haven't. This makes the eBay user experience less friendly and more frustrating.
So I think eBay should offer its own (free) sniping software built right into the site - give bidders the opportunity to make a public or a private bid (or both). Only the highest public bid is shown but the highest bid overall (public or private) will actually win.
Why would anyone make a public bid if a private one is available? Wouldn't this turn all eBay auctions into silent (?) auctions? Not necessarily - there are some circumstances when a public bid is actually in the bidder's favour - for example if there are several auctions for equivalent items all ending around the same time, making a public bid on one of them is likely to push other bidders towards the other auctions.
Though that bit of game-theoretic oddness could also be eliminated with a closely feature closely related to private bids, which is the ability to (automatically) withdraw a private bid. This would allow one to bid on several auctions at once, while guaranteeing that you'll only win one of them. More complicated logic would also be possible, like "I want to buy either A or the combination of (B and any of C, D or E)". I'm not sure if this is currently possible with sniping software (I haven't used it). One could also set different bids for different auctions, if some are more desirable than others in some ways.
All these changes favor buyers rather than sellers, so eBay users who are primarily sellers probably wouldn't like them (after all, they help buyers save money!) But sellers already hate eBay - many of their other policies are drastically biased towards buyers. The only reason that sellers keep selling stuff on eBay is that is where the buyers are (and therefore that is where the best prices are, even after factoring out eBay's cut).
One other reason that eBay might want to do this would be that by having private bids go through the site, they get more accurate information about who is prepared to pay how much for what. I don't know if eBay currently does anything with this sort of information, but it surely must have some value to someone.