Some tissues such as liver are particularly difficult to work with. Only rarely that you can get an RIN value out of it, let alone 6.5. I find QIAGEN RNeasy kit most effective in extracting rna from liver.

Instead of RIN value, you can assess your rna extracts by their purity (260/280, 260/230) and integrity of bands (28s, 18s) on gel. You can always mention these detail on your methods, and those who have experience working with liver should understand. (Although it's always best to try with few samples first.) Good luck!

What has the concentration of the NA samples? Make sure it's sufficient to get a good reading. What did the virtual gel look like? Was it obviously bad, or was there something strange in there? Feel free to post a couple of pics here of the best vs an NA sample.

RIN values may not be calculated for several reasons, like when an unexpected peak is detected, and errors can be considered either critical or non-critical depending on the type. I've attached a document that may help you, if you need it.

You mentioned the TissueLyser. In my experience that is a good way to thoroughly homogenise your samples. What method of homogenisation did you use?

Also think about your sample collection and storage. Could some not be optimal there?

If you have the chance to do some kind of column extraction as opposed to Trizol then it's worth testing a few samples. I've done many many (fish) liver samples with a GE Health column-based kit, and another Qiagen kit with really good results (RIN values typically >8) from RNAlater preserved tissue.