I'm building a bike for my girlfriend and she's on the shorter side (5'3)

She currently has a MTB with a 12" BB height with 26" tires but its not working because to get her saddle at the proper height she then feels like its too hard to get her feet on the ground at a stop.
This bike will be used only on the streets so ground clearance isn't a big deal

Any frames I can look into that have a low BB height?
Also planning on using 165mm crank arms (she has 170 right now)

You mean with both feet on the ground or something? That's a pretty low BB regardless of leg length. Maybe a touring bike would work, but the vast majority of mtn and road bikes will feel too high if she wants both feet on the ground. My wife has stubby legs [I hope she doesn't see this], and can't touch both feet on the ground. She leans over and gets a toe down. It's just something you get used to...

Electra Townie or any Rivendell frame.

Most touring frames have a low BB. The Surly LHT feels especially low to me.

I'm short legged. I have a 1970s Coppi steel road bike with a slightly less than 10" BB height. The frame size measures out at 50.5 cm. On another subforum here someone said a while back that Italian road bikes are good for short legged riders. I don't know if that's true in general but it just so happened that I found a good fit with an old Italian road bike.

FYI, she's not supposed to be able to touch the ground while remaining in the saddle.

Top tube standover height determines if a bike is too tall for your legs, and with a women's mountain bike that likely has the sloped top tube, she should have no problem touching the ground.

The seat should be adjusted so that the leg is almost fully extended when the pedal's furthest away, like the 5-o'clock position. The knee should be darn near straight at full extension, but the hips should not sway while riding. If configured properly, no normal mountain or road bike will allow you to comfortably sit in the saddle while touching the ground. If a seat is too low, knee problems will ensue, and the inefficiency of each pedal stroke will make for a lot more effort (harder to push) than needs to be made.

If that's what she's looking for, check out a Giant/Revive, recumbent bikes, or some of the crank-forward comfort-cruisers. Those bikes have the kind of geometry where the seat can be adjusted properly while still remaining close enough to the ground to allow planting both feet.

Actually with the wonderful Electra Townies you are supposed to be able to place both feet on the ground while sitting.

Perhaps you missed this last part of my post, but I mentioned Crank-forward bikes, a class in which the Townie is definitely included. It's not as pronounced as the Rans Fusion or the GT Kustom Kruiser bikes, but you're right, it's an excellent bike for someone who is looking for riding geometry suitable for flat-footing while in the saddle.

Still, it's not accomplished merely by placing the BB closer to the ground, it has more to do with putting the crank out further in front of you than on a hybrid or road bike, so that the arc your hip joint forms will allow your feet to touch the ground, without having the seat too low.