Someone sent me a friend request. I didn't know who they were because their Facebook profile was obfuscated, so I rejected it. Later, they told me they were the one who sent it. So, I asked her to send the friend request again, but there was no option to do so. There appears to be no option for me to send one, either. We are stuck only able to send messages.
Is there really no way to "reset" the situation and become Facebook friends?
According to this it seems you have missed the chance - until Facebook start paying some attention to users, so probably a very long wait.
Here has:
but they will be able to send you another friend request in the future.
There seems to be general uncertainty about how far into the future. I have seen estimates of six months, eight to twelve months and a couple of more firm sounding "1 year"s, so I would assume one year.
If you have set Who Can Follow Me (Settings -> Public Posts) as Public, and you delete the request (but didn't mark it as spam), you can see that person in the Followers section under Friends on your profile.
Click on the name of that person and you will see Add Friend button on their profile.
According to https://www.facebook.com/help/131351653607683 you will have to wait one year before being able to send another request.
