Is the word "Allah" appearing on a prayer rug considered a sign or good tidings from God to my friend to strengthen her in her prayers and hijab after her repentance, especially since she no longer feels suffocated when wearing it?

Your friend must make up all the prayers she missed after reaching puberty. If she doesn't know the exact number, she should continue making them up until she is reasonably certain that she has absolved herself. As for her seeing the word "Allah" on the prayer rug, that might be a warning to her about the mistake of not wearing the hijab (definition).