With the advent of CRIME, BEAST's successor, what possible protection is available for an individual and/or system owner in order to protect themselves and their users against this new attack on TLS?
This attack is supposed to be presented 10 days from now, but my guess is that they use compression.
SSL/TLS optionally supports data compression. In the ClientHello message, the client states the list of compression algorithms that it knows of, and the server responds, in the ServerHello, with the compression algorithm that will be used. Compression algorithms are specified by one-byte identifiers, and TLS 1.2 (RFC 5246) defines only the null compression method (i.e. no compression at all). Other documents specify compression methods, in particular RFC 3749 which defines compression method 1, based on DEFLATE, the LZ77-derivative which is at the core of the GZip format and also modern Zip archives. When compression is used, it is applied on all the transferred data, as a long stream. In particular, when used with HTTPS, compression is applied on all the successive HTTP requests in the stream, header included. DEFLATE works by locating repeated subsequences of bytes.
Suppose that the attacker uses some JavaScript code which can send arbitrary requests to a target site (e.g. a bank) and runs on the attacked machine; the browser will send these requests with the user's cookie for that bank -- the cookie value that the attacker is after. Also, let's suppose that the attacker can observe the traffic between the user's machine and the bank (plausibly, the attacker has access to the same LAN of Wi-Fi hotspot than the victim; or he has hijacked a router somewhere on the path, possibly close to the bank server).
For this example, we suppose that the cookie in each HTTP request looks like this:

Cookie: secret=7xc89f+94/wa

The attacker knows the Cookie: secret= part and wishes to obtain the secret value. So he instructs his JavaScript code to issue a request containing in the body the sequence Cookie: secret=0. The HTTP request will look like this:
POST / HTTP/1.1
Host: thebankserver.com
(...)
Cookie: secret=7xc89f+94/wa
(...)

Cookie: secret=0

When DEFLATE sees that, it will recognize the repeated Cookie: secret= sequence and represent the second instance with a very short token (one which states "previous sequence has length 15 and was located n bytes in the past); DEFLATE will have to emit an extra token for the '0'.
The request goes to the server. From the outside, the eavesdropping part of the attacker sees an opaque blob (SSL encrypts the data) but he can see the blob length (with byte granularity when the connection uses RC4; with block ciphers there is a bit of padding, but the attacker can adjust the contents of his requests so that he may phase with block boundaries, so, in practice, the attacker can know the length of the compressed request).
Now, the attacker tries again, with Cookie: secret=1 in the request body. Then, Cookie: secret=2, and so on. All these requests will compress to the same size (almost -- there are subtleties with Huffman codes as used in DEFLATE), except the one which contains Cookie: secret=7, which compresses better (16 bytes of repeated subsequence instead of 15), and thus will be shorter. The attacker sees that. Therefore, in a few dozen requests, the attacker has guessed the first byte of the secret value.
He then just has to repeat the process (Cookie: secret=70, Cookie: secret=71, and so on) and obtain, byte by byte, the complete secret.

What I describe above is what I thought of when I read the article, which talks about "information leak" from an "optional feature". I cannot know for sure that what will be published as the CRIME attack is really based upon compression. However, I do not see how the attack on compression cannot work. Therefore, regardless of whether CRIME turns out to abuse compression or be something completely different, you should turn off compression support from your client (or your server).
Note that I am talking about compression at the SSL level. HTTP also includes optional compression, but this one applies only to the body of the requests and responses, not the header, and thus does not cover the Cookie: header line. HTTP-level compression is fine.
(It is a shame to have to remove SSL compression, because it is very useful to lower bandwidth requirements, especially when a site contains many small pictures or is Ajax-heavy with many small requests, all beginning with extremely similar versions of a mammoth HTTP header. It would be better if the security model of JavaScript was fixed to prevent malicious code from sending arbitrary requests to a bank server; I am not sure it is easy, though.)

Edit 2012/09/12: The attack above can be optimized a bit by doing a dichotomy. Imagine that the secret value is in Base64, i.e. there are 64 possible values for each unknown character. The attacker can make a request containing 32 copies of Cookie: secret=X (for 32 variants of the X character). If one of them matches the actual cookie, the total compressed length with be shorter than otherwise. Once the attacker knows which half of his alphabet the unknown byte is part of, he can try again with a 16/16 split, and so on. In 6 requests, this homes in the unknown byte value (because 26 = 64). If the secret value is in hexadecimal, the 6 requests become 4 requests (24 = 16). Dichotomy explains this recent twit of Juliano Rizzo.

Edit 2012/09/13: IT IS CONFIRMED. The CRIME attack abuses compression, in a way similar to what is explained above. The actual "body" in which the attacker inserts presumed copies of the cookie can actually be the path in a simple request which can be triggered by a most basic <img> tag; no need for fancy exploits of the same-origin-policy.
