29 yr female. Best option for weak jaw/recessive chin? (Photo)

You are not AT ALL a case for liposuction at the neck, absolutely not. All the patients with weak chins have
flaccid submental "hammock" due to the short distance of attachment of
the floor-of-the-mouth musculature; this flaccid submental area is
authomatically corrected after the OSSEOUS genioplasty, NEVER with chin
implants.
I perform regularly all kinds of genioplasties, not only sliding
genioplasties but also one, two and three-dimensional augmentation
and/or reduction genioplasties, plus other custom designs; in other
words, I know well what I am talking about.
You are a case for a sliding genioplasty to increase onlychin projection, or alternatively performing a two-dimensional augmentation genioplasty to increase vertically as well as anteriorly (to be discussed) your chin dimensions.
This is an awesome, lifetime lasting, stable and safe procedure, which
notwithstanding needs very accurate preoperative antropometric
planning, desing, discussion and a brilliant execution, a technique full
of creativity and good aesthetic judgement requirements.
Chin implants have many disadvantages:
-visibility, static or with facial gestures / smiliing
-artificialness, fake look, evidence of implant presence

-limited lack of versatility, they do not provice vertical modifications neither reductions
-common infections
-common dislocations
-possible nerve irritation and pain
-short lifetime
So sooner than later they are normally removed, or exchanged by an osseous genioplasty.
Why
so popular (chin implants) then? Simple answer: they are technically
accessible to all surgeons, little training is required, can be done
under local and sometimes office based procedures, they are cheap.
Osseous
geniplasties are more expensive and need good skills, true, but when
performed they produce lifetime results, safe, stable overtime, very low
rate of complications, versatile three-dimensional modifications
(augmentation or reduction, vertical, horizontal and depth), low rate of
reinterventions, etc.
Feel free to request any additional information from me.

You have both nonsurgical and surgical options:
If you want to see how added chin projection can help your profile a good nonsurgical alternative is Voluma, a dermal filler used to build up bony prominences like the cheeks but can also be used to excellent effect in the chin. In fact, many patient who come in to discuss chin implants (which require surgery) will opt first for a Voluma chin augmentation just to see if they like the look. The product is very safe and lasts 2 years or more. You could always proceed to the permanent option (the implant) in the future, or just continue the filler. 
I would recommend combining this with refinement of your neckline by eliminating submental fat (the "double chin"), either by liposuction (which is technically a surgery, though minimally-invasive) or with Kybella, an injectable that can achieve similar results in appropriate patients without surgery. See the video below. I have a Youtube video demonstrating Voluma injections as well, which can be easily found by searching on my name.These nonsurgical procedures can be performed at the same time, during the same visit, if you wish.

Best wishes!  Harry V. Wright MD, Sarasota, Florida