I'm hoping someone can give me some advice on what to say to my neighbor. She is in her 60's and I just learned that her DH, who was awaiting a kidney transplant, passed away on Friday. We don't know each other very well, but we are friendly and I want to be able to say or do something for her.

Any suggestions?

I think most people just like you to acknowledge & give them a chance to talk if they want.
You could say you heard about her husband and express your sympathy. She may just talk or cry or whatever.
If you feel comfortable, you could offer some help or something. I know what is nice is if you made some healthy food that is easy to re-heat (like soup or something). When my dad passed away, none of us felt like eating & even months later, my mom still didn't feel like cooking & it was nice to have food that she could just defrost & eat.

I agree with the above post.

Let them talk, and offer to help.

Acknowledge the person's grief, the deceased person's life and passing, and offer to help with a meal or a small task.