Internet weather device using wifi?
The Twatch is the right kind of device. But, it's wired ethernet only.

The firmware will need some tweaks - currently it connects to twitter's API over TCP/HTTP and parses JSON.





The parts list doesn't include prices, but the BOM should easily be under $30 in quantity.

Part    Value   Package
IC1     PIC 18F67J60    TQFP-64
C1-3    10uF tantalum capacitor, 10volts+   SMC_A
C4,5    33pF capacitor  0805
C10,11,C14-23   0.1uF capacitor     0805
ICSP    5x 0.1″ male pin header     
J1  2.1mm SMD power jack    SMD
J2  HR911105A ethernet jack     RJ-45
L1, L2  Ferrite bead, 200ma+    0805
NPN1    NPN transistor, 250hfe+, 800ma+     SOT-23
Q1  25MHz SMD crystal   HC49UP
R2(A)   10K single turn trim resistor   3mm SMD or through hole (cheapest)
R3  240 ohms resistor   0805
R4-6    390 ohms resistor   0805
R7  2,260 ohms resistor, 1%     0805
R10-21  10,000 ohms resistor    0805
R30-33  49.9 ohms resistor, 1%  0805
VR1     LDO 3.3volt regulator (LD1117)  TO-220
VR2     7805T 5volt regulator   TO-220
HD44780-LCD     20×4 HD44780 character LCD