Hey Jim,
Really you should look into reframicating the software if only that others may use it on UNO's buuuut,
Check out the Teensy++!
Teensy USB Development Board
128k flash, 8k SRAM, 4k EEPROM.
46 I/O pins; 8 analog inputs, 9 PWM's.
USB based Atmel chip, programmable as HID/MIDI and they have libraries.
Really small. Arduino compatible, able to use the Arduino IDE.
Way more than an UNO, not sure how it rates compared to a MEGA2560.
With pins (sticks right into a breadboard), $27. Without, $24.
Or you can buy SPI-bus SRAM, 8k x 8-bit through-hole DIPs are pretty cheap @ $.70. See the 3K640-I/P:
RAM's (Random Access Memory ICs)
Not as fast as on-board SRAM but way cheaper than a new board.
The 23K256-I/P 32k x 8-bit SPI-bus SRAM are surface mount, for $1.35 they are 4x as much.
Of course either way there's a shipping charge so save up and buy enough to justify that.
And my non-disclaimer is that I have no connection to either of those companies except being a satisfied customer. PJRC (the Teensy site) shipping got to me a lot quicker but then across the US isn't as far from me than Hong Kong. But the HK shipment 'only' took between 2-3 weeks and they DO have so many different things.
There are many other good sites but those jump out at me when it comes to 'Mo RAM!'. Hey... Moe RAM, that'd be a good forum name, wouldn't it? People could say "Hey Moe!".