Your desired spec is a little overpowered and certainly you're not going to get it for less than $30. A 50MHz CPU is waay over the top for a weather readout given that you can connect a 16MHz chip up to Ethernet just as easily. Similarly, a 5" LCD is probably going to run you at least $20 anyway.
If you want 'WiFi', you could cut corners and use a base device plugged in via Ethernet. This would then talk to another module connected to the screen (wherever you want to place it). Ethernet and 2.4GHz transcievers (e.g. Nordic nrf24l01+) are comparatively cheap compared to fully fledged Wifi modules if you build all the circuits from scratch.
You can use a cheap Ethernet chip like the ENC28J60 to connect to the router, you'd then need to either find code or write your own TCP/IP stack. Google yields some application notes, and here is a web server written for an Atmega168. http://en.klinkerstein.m-faq.de/index.php/MicroWebServer. Personally I think you'd be best off just building it from scratch using a bare AVR and an Ethernet controller rather than paying for an Arduino which will negate the point of saving money.
For the receiving end, you'd need another wireless transciever (2.4GHz or radio, your pick), another microcontroller - say an Atmega168 for arguments' sake - which recieves the weather information and displays it. Using a bog standard HD44780 16x4 LCD you could display more than enough information. If you need graphics, pick up a KS108 based screen. Plenty of code and drivers on the web for LCDs.
I would hard code the format into the display micro and when you recieve a weather update, store it in EEPROM. All you'd need is, say for 5 days, 5 values for temp, wind, weather type, etc. 1kB is more than enough space. EEPROM does have a finite r/w limit, but realistically if you update the weather 24 times a day you could still go for years.
You'll probably need to go surface mount to be seriously cheap, and as for a BOM, it's hard to list one without an exact design spec. I would say you'd be looking at around $40 maximum - bear in mind that's less than the cost of some Wifi modules anyway.
NRF2401L (QFN) - $5 each
ATMega168 - $3 each
16x2 LCD - $5 on eBay
Other components like resistors, caps and so on are negligible compared to the above. You might have to pay a dollar or two for the Ethernet plug and the antennae for the transceivers. Prices were checked on Farnell UK, converted roughly into dollars.