Congrats on setting your mission!
There are several different ways where you can work on a project guitar.
1) Revive an old dead guitar
2) Upgrade and make over an off the shelf guitar
3) Retrofit an off the shelf guitar - rerouting, changing necks etc..
4) Customise assembly of guitar parts
5) design and build one from raw materials
Yes, wood type matters. But the difference is significant only when it is well assembled and setup.
Assuming playability will not be compromised, What are you after?
1) Looks?
2) Function?
3) Replica of an Artist Model?
4) Artistic Expression?
5) Unique tone?
6) One of a Kind model?
7) Some of the Above?
8) None of the Above?
A good first project would be to bring a dead guitar back to life. It is a challenge because you'll be torn between keeping the original specs and setting on a trial and error adventure.
If you're looking at custom assembly of a new guitar and willing to start from unfinished parts, $500 may be enough. I've personally put together some project guitars which cost me much less than that (minus labour cost, electric bill, tools, oils & finishes that is). Brands are not everything. It is very easy to be swayed to buy parts only from Big name manufacturers but it doesn't really mean that you'll get a better guitar afterwards.
Visit
www.projectguitar.com for a start... you may get more ideas there...
God Luck!