Option 1 is not really possible because he already asked her out and got rejected. So there is no uncertainty here for him to live in. There is only denial.
So his choices are really
1. Live in denial.
2. Ask her out again, to confirm she hasn't changed her mind.
At this point I'd just encourage him to go with number 2, since he's obviously not taking our advice about letting it go.
@Slothbroth I'm officially recommending that you become "that guy" that asks her out all the time. Just make a joke about it. Like "first you won't go out with me, and now you get my order wrong? It's like you enjoy breaking my heart!" You know that kind of thing.
Or ask her advice. "This lady at work asked me out. I'm not sure if I should say yes, what do you think?" And show her a picture. Then you can joke and say "hey don't get jealous, you had your chance. YOU rejected ME, remember? Did you think I was gonna wait around forever for you to change your mind?"