The LG G2 isn't exactly one of the smaller phones around, and I think LG knew that from the beginning, too. Although they've tried to make the bezels as small as possible, the phone itself with its 5.2" screen still ended up a little too big for most people.
A new rumor says that LG may be following in Sony's footsteps and release a "mini" version (although it may not use the name Mini), that will have almost the same specs as its bigger brother, but with only a 4.7" screen. That's still not exactly "mini", but if it has bezels that are just as small as the LG G2, then it should be roughly about as big as a Moto X, which fits quite nicely in most people's hands, while still having a large enough screen.
As mentioned, the phone will have the same Snapdragon 800 processor as the G2, but it's unclear yet whether it will have a 1080p resolution or a 720p one. I'd probably bet on it having 720p, since they won't be able to fit a battery as large as the one in the G2, unless LG decided that it's better to compromise on battery life, but have identical specs otherwise (except the battery size/battery life). It should also come with 2 GB of RAM.
I wouldn't expect anything less than 16 GB of storage for the base model, especially since I believe all flagships should come out with 32 GB of storage next year. At this point it costs OEM's much less to put 32 GB of storage in their phones, than it did to put 16GB inside, a few years ago. It's just that they've decided not to do it, either to push other specs instead, or take in that extra profit. This is the "mini" version, though, so they might want to make it slightly cheaper. In that case, I'd understand why they'd opt for 16 GB.
There's no potential release date yet, but the phone could be announced at CES about 4 weeks from now. Saving it until MWC might not make the phone as exciting, with all the new next-gen flagships being announced around that time, from other OEM's, so a CES announcement seems pretty likely.