Solvent for caffeine extraction
Following is the list of solvents previously checked for the solubility (From reference 1 below):

$$
\begin{array}{|c|c|c|c|c|c|}
\hline
\textrm{Solvent} &amp; \textrm{Temperature, } \pu{^{\circ}C}  &amp; \textrm{Solubility, } \pu{\%} (w/w) \\ \hline
 \textrm{Water} &amp; 25 &amp; 2.2  \\ \hline
 \textrm{water} &amp; 100  &amp; 66.7  \\ \hline
 \textrm{Ethanol} &amp; 25 &amp; 1.2 \\ \hline
\textrm{ Ethanol} &amp; 60 &amp; 4.5 \\ \hline
\textrm{Ether} &amp; 25 &amp; 0.3 \\ \hline
\textrm{Ethyl Acetete} &amp; 25 &amp; 2.5 \\ \hline
\textrm{Chloroform} &amp; 25 &amp; 18.0 \\ \hline
\textrm{Acetone} &amp; 25 &amp; 2.0 \\ \hline
\textrm{Benzene} &amp; 25 &amp; 1.0 \\ \hline
\end{array}
$$

My suggestion is to use ethanol in $\pu{60 ^{\circ}C}$. However, I'm not sure how ethanol would behave with the presence of tannins. For example, in a separate literature report, $\pu{1.0 mg}$ of tannin was isolated from a $\pu{g}$ of grape skins using 100% ethanol while the amount was increased to about $\pu{5.5 mg}$ when solvent was switched to 100% acetone (Ref.2). 

References:


Q. V. Vuong, P. D. Roach, “Caffeine in Green Tea: Its Removal and Isolation,” Separation &amp; Purification Reviews 2014, 43(2), 155-174 (DOI: 10.1080/15422119.2013.771127).
M. O. Downey, R. L. Hanlin, “Comparison of Ethanol and Acetone Mixtures for Extraction of Condensed Tannin from Grape Skin,” South African Journal for Enology and Viticulture 2010, 31(2), 154–159 (DOI: https://doi.org/10.21548/31-2-1412).