Well Becky, I vote for "Sue." That was the very first name to spring to my own mind, instantly and quite unbidden, long before I saw that other like-minded posters had also proposed it.
Mind you, I like Ann(e) as well, just as several other posters do here. I can't call this "original," but I guess I like "classic" names. They combine dignity with femininity. In addition, with a two- or three-syllable first name, I felt it was best to pick a simple, shorter sounding middle name. "Becky Ann(e)" sounds fine, but the only problem is when we follow "Rebecca" with a name beginning with a vowel. Then we end up with a glottal stop in between, a sort of verbal "hiccup." That's unless speakers interpose a consonant in between that isn't really there, the way many people talk about "vanillaRice cream," for instance. "Rebecca Sue" on the other hand "flows" just as gracefully and euphoniously as "Becky Sue" does.
Besides, it was inevitable that "Sue" would come to mind, because it sounds so completely natural:
"If you knew
Becky Sue
Then you'd know why I feel blue
Without Becky
My Becky Sue..."
Didn't somebody sing something just like that once? Long ago it was, "before the music died..."
All the best, whatever you choose!