Does relative amount of lithium and cobalt oxide change in recharged battery? There is a question on my assignment about stoichiometry. I spent 2 whole days searching and review this topic but still clueless.

Reaction in lithium ion battery
$$
\ce{Li(s) + CoO2(s) → LiCoO2(s)} \, .
$$
When the battery is recharged, electrical energy will reverse the reaction. How do the relatives amounts of lithium and cobalt oxide change as the battery is recharged and how?
About maximum $20~\%$ (as I remember, perhaps more now) of the $\ce{Li}$ goes (as $\ce{Li+}$) out of the structure of $\ce{LiCoO2}$, through the elctrolyte, and into the electrode on the other side of the battery. That electrode consists mainly of graphite, which intercalates the $\ce{Li}$ in it's layers in a quasi-metallic form.

So it is $\ce{Li_{1-x}CoO2 + x~Li}$ after charging. $\ce{LiCoO2}$ has a special structure that does not break down when some $\ce{Li}$ is missing, except if you keep a charged battery at over $\mathrm{50~^\circ C}$ or so.

By the way, the related wikipedia article tells you all this. Two days of researching? ;-)