I have the AutoFill option enabled in my Safari settings, and I am not doing Private Browsing. However, Safari will not offer to save my Ebay password no matter how many times I log in. I have also tried clearing cookies and history. Anyone have any ideas?
The eBay mobile site requests Safari not to have passwords saved, and Safari honors it.
eBay is not the only site that behaves like this, PayPal is another prominent example. Many online banking sites (that I know of) prevent autofill too.
On a desktop browser you can get around this limitation with an extension (see for example https://discussions.apple.com/thread/4160095?start=15&tstart=0). I don't think you can force your iPhone to ignore it. You can download eBay's app, though.
The technical explanation is that the username field in the login page sets autocomplete to off and so prevents autofill:
To see this fragment of HTML I changed my browser's user agent to Safari iOS 5.1 (I use Safari 6 on OS X):
loaded https://signin.mobileweb.ebay.com/login?mfs=ftr, showed the page source and searched for