An attempt at a brief summary of solar charging ICs for single-cell lithium batteries.

Solar panels provide the most power if you limit the current drawn from them, so a proper solar charger will do MPPT (Maximum Power Point Tracking). In reality, most cheap solar charger ICs imitate this by restricting the current drawn to keep the panel voltage constant (generally around 4.5V).

Best Bets

You probably want to use:

Consonance CN3083 or CN3082 - it seems, confusingly, that the CN3082 is the later model (available from ebay), ~£1 each. SOP8 package. Drawback: maximum 6V input voltage will necessitate either a high-power zener or a LDO regulator to protect the input for most solar panels.

Other Options

Linear LT3652, £4-5, 40V max input,

Packages probably unsuitable for hand-assembly

TI BQ24040, <£1 each, unimaginably small 10WSON package, otherwise ideal.
Linear LTC4121, ~£3 each, configurable maximum power point, QFN16 package.
Linear LTC4079 ~£3 each, 60V max input, DFN10 package.

Table

MFN	MFP	Max U (abs)	Max I	Pkg	Price/ea	Source	Datasheet
Consonance	CN3083	6V (6.5V)	600mA	SOP8	~£1	Ebay	[PDF]
Consonance	CN3082	6V (6.5V)	600mA	SOP8+EP	~£0.5	AliExpress	[PDF]
~~Microchip~~	~~MCP73831~~	6V (7V)	500mA	SOT-23	~£0.4	DigiKey	[PDF]
Linear	LT3652	32V (40V)	2A	12-TSSOP+EP	£4-5	DigiKey	[PDF]
TI	BQ24040	6.8V (32V)	1A	10-WFDFN+EP	~£1	DigiKey	[PDF]
Linear	LTC4121	40V (43V)	400mA	16-QFN+EP	~£4	DigiKey	[PDF]
Linear	LTC4079	60V (62V)	250mA	10-DFN+EP	~£4	DigiKey	[PDF]

~~Not a solar charger (USB or simmilar)~~