Solar cell charging circuit
I'm not sure yet exactly, but say a standard, 2v, 20mA 5mm led


So, 2 V · 20 mA = 40 mW power.


  maybe get about 15 minutes 


ok, let's round that up to 25 minutes:

25 min · 60 s/min · 40 mW = 60 J

is your energy storage requirement.


  Is it possible to use this a 1v solar cell to charge a battery? Maybe by boosting the voltage?


yes.


  What kind of energy storage would be good: Lipo, NiMH, capacitor?


With the requirements above: probably simply a supercapacitor.

Let's say you charge that up to 5V. The energy in a 1 F supercap then is

W = 0.5 · C · U² = 0.5 F · 25 V² = 12.5 J

so, about twice as much as we strictly need, which is nice, since it gives us headroom for efficiency. You could of course use larger storage, which would give you more on-time :)

Also, supercaps have a high self-discharge rate, so this will only work if you want light shortly after the sun has gone down. If you want light hours later, better use a lithium or NiMH battery. These will have capacities that will let your LED run for hours, if not days. However, they might require more effort to charge. See the paragraph on dedicated ICs below.


  Are there circuit examples I can follow


Certainly!


  Can these circuits build in a cheap way?


Yes, and that answers the previous question, too: There's a lot of ICs designed to make charging of supercaps or batteries from small solar cells easy. I remember TI having a range of interesting chips, but I'm sure Maxim has some, as well as others, too. Their Datasheets and application notes will have good examples! Maybe, they even have simple eval boards. Or you can buy such pre-made on the internet.

Go into the power management / power harvesting category of these sites and look for solar charger circuits.

To drive the LED from the charged capacitor most efficiently, you'd probably use a switch-mode converter, too.