If I were you, I would definitely -not- go the route of wiring up a keyboard; depending on how well you want it to work (and how you want it to react), it could eat up your entire budge. Instead, I would visit a local thrift/second-hand store, and see if I could find an old cheapo synthesizer. Depending on the item and where you find it, and how well it still works, you shouldn't have to spend more than $20.00 (unless it is a larger-size synth).
Make sure whatever one you find -doesn't- have MIDI IN/OUT - why? Because those keyboards, if they work, should be reserved for the budding (or poor) MIDI artists out there. Also, research the heck out of whatever cheapo synth you buy; there are a few out there that are considered "classic" (especially among bender crowd - but also among collectors), and if you tear one of those up, you will only end up crying when you find it is worth waaaay more than what you paid for it.
You might pick up a couple to expand the available octave range if the ones you find are of the smaller variety (and if you can find two or more the same size; or go tiered?); what you won't find on these keyboards, though, is advanced features like velocity control and other such things you find on better synthesizers (but that's OK - you couldn't build such things into a playable home-brew keyboard anyway for $100.00).
Once you have your cheapo-synth, and you know it isn't anything anyone cares about (or maybe you don't care; just realize, though that such an act of "restruction" would be considered by some akin to taking a crayon to an original copy of the Magna Carta), then you at least have a pre-made keyboard, plus a lot of other parts you could use in your custom synthesizer (including a sound chip and amplifier, if you care to interface to them!).
Oh - did I mention "Circuit Bending"? You may want to look into that scene; while in some cases they have done the above "sacrilege" against certain keyboards and electronics (especially speak-n-spells; then again, there were a ton made, so whatever), some of their techniques might be of interest to you.
I generally agree with Grumpy_Mike, and he has educated me on some things (and maybe he will here, too) - but I don't think sound generation is outside the scope of an Arduino. I have nothing to back this up (its on my "someday" list), but I think the Arduino would be more than capable of generating 8-bit single channel FM, or possibly 12-bit dual-channel FM (6 bits per channel), using simplified wave tables and ASDR envelopes (DDS - direct digital synthesis). Depending on how well you write the code, you might be able to have multiple voices mixed on the output. Output of the sound could happen either via a DAC (a simple resistor ladder DAC would be fine); you could also possibly output the sound using PWM and filtering.
What it couldn't do, though, is all of that plus keyboard scanning; I do think it could act as the synthesizer sound source, alone, though. Alternatively (and perhaps within your budget as well), you could interface with a sound chip like this one:
You might also find specs on-line for the sound chip in the original synthesizer; you have a number of options available ultimately.
Good luck with the project; take it in simple, easily digestible chunks and steps, and it should work out!