The answer is to use a charging chip that has the ability to back off on charging current when the input voltage from the solar panel begins to collapse.

While using a regular Li-Ion charging IC will technically "work" with a solar panel, it will take forever to charge in practice, and, as mkeith pointed out: "it will never get a chance to terminate charging".

The discussion about this on the Adafruit website is the best explanation of this phenomenon that I can recommend to others with this question:

Most people try to plug a solar panel directly into a lipo charger and while it sort of works, the battery takes forever to charge because the efficiency is terrible! That's because most lipo chargers are meant to plug into a USB port or wall, and are very simple in their design. USB ports supply 5V at up to 500mA and they're pretty solid - the voltage doesn't change much even at the max current draw. So when you plug a charger into a computer with a USB port, they just draw 500mA or so and happily chug away. Same goes for wall adapters. The voltage and current limits are kept steady. Solar panels are a little different, the voltage and current vary constantly depending on sunlight available. They are unstable! That instability confuses battery chargers, which causes them to do one of two things: rapidly turn on and off as they try to draw more current from the panel than possible and/or draw much less current than they can, to keep the voltage from collapsing

Time Bomb

The true problem is the invisible damage that happens inside the cell, producing a few atoms short of a major short-circuit, until a little vibration, jarring, or a dropped flashlight, moves it the rest of the way, completes the circuit, and you find out you're holding a roman candle.

Active Cutoff Needed

In order for this flashlight to be safe with this battery, it would have to have a more active cutoff, because without it, you will eventually reach the 1.5 voltage point, and then this happens:

Copper Shunting

At battery university, at "BU-808a: How to Awaken a Sleeping Li-ion", the following section can be found:

Do not boost lithium-based batteries back to life that have dwelled below 1.5V/cell for a week or longer. Copper shunts may have formed inside the cells that can lead to a partial or total electrical short. When recharging, such a cell might become unstable, causing excessive heat or show other anomalies.

Venting With Flame!

Then, also at Battery University, on the page, "BU-304b: Making Lithium-ion Safe", it contains this little gem of what might be considered a good example of ultra-cheap-manufacturing double-speakery:

Manufacturers of lithium-ion batteries do not mention the word “explosion” but refer to “venting with flame” or “rapid disassembly.” Although seen as a slower and more controlled process than explosion, venting with flame or rapid disassembly can nevertheless be violent and inflict injury to those in close proximity.

The Longer I Think About This...

So, what happens if you charge the over-discharged battery, and it has shunts, but they smolder, or otherwise don't actually cause a problem until you've fully charged the battery, and screwed it up tight in the flashlight. Do you now have a grenade? I just wouldn't play with this. Convert it to use protected-cells, if you just can't let it go to waste. Let's say somebody else accidentally gets this flashlight? (a thief breaks in, and takes your flashlight for his little girl). Don't let it happen... I know... Put it through your own version of the inquisition: "convert, or die!" ;-)

Discharged == Harmless

In a talk by Professor Jeff Dahn of Dalhousie University entitled "Why do Li-ion Batteries die, and how to improve the situation", we find out the following things: Over-discharged cells are not dangerous until you recharge them. When Lithium ion cells are made, apparently, their ingredients are relatively safe, because Jeff says that you can assemble them in the open air. But when you charge them, both anode and cathode (and it's a jellyroll, so you have plenty) become very reactive.

So, then, based on Jeff's talk, if a Lithium ion cell is fully discharged, then it is no longer dangerous, but should at that point no longer be used, but recycled. I run mine down to zero volts, then I put them in a group to be recycled together. If somebody else should throw them away, they shouldn't cause a fireworks show in the trash truck while it's compacting because they have no stored energy, and there is no longer the highly reactive intercalated Lithium.

Charging Only to 4.1 Volts is Best

In the same video, which I consider to be some of the most authoritative information on the subject (I've watched the video many times for that reason), Jeff says a few things which stick in my mind and I want to pass on:

1. "Time of exposure to high temperature while charging or discharging is the bad thing." He's talking about the electric cars and some real problems some of the manufacturers have gotten themselves into, most notably, one manufacturer who decided NOT to thermally manage their cells. (Think summer in Arizona, and discharging cells to drive a car, which produces a LOT more heat, and bad chemical reactions that quickly kill the cells under those conditions.)

2. If you're storing them, do not store them fully charged. Discharge them so they only have a little left. And store them in a cool place, but not in the freezer. Refrigerator or basement would be good. For me, this is hard to do, so my compromize is to charge them just before I use them, with a little cool-down time in-between. And I try not to leave my rechargeables in the car!

3. Charge to only 4.1 volts. (Found this in the Q&A at the end of the video). Then, in Jeff's words, "if they put the cell in your body, they'll last until you're dead... or until I'm dead..." Made his inquisitor speechless.