ROM hacking to make yourself a new game is generally considered a backwards way of setting about things, ROM hacking to do this... probably even more so.
One tends not to overwrite cartridges, the Read Only Memory part of ROM speaking to that one.
What you probably want is instead a homebrew audio player on some flash carts. Commercial N64 flash carts are expensive but as you presumably don't need much for this you could probably get away with a rather more minimal affair. I don't know much about the N64 for this sort of thing but have seen it before for the GB, GBC, GBA and DS, indeed http://www.littlesounddj.com/lsd/
for the first of those mentioned is fairly highly regarded and you get things like http://reinerziegler.de/readplus.htm#Home%20made%20carts
(see also the GBA page) to support some of them -- none of those are great for those looking to wander around with the console's entire library in their pocket but for a single program they are just fine, though in the case of the GBA and DS you can probably pick up suitable commercial flash carts for next to nothing.
There are plenty of people which do audio creation for old consoles. The NSF scene for the NES and the more general demoscene providing ample example here. The focus there tends to be more pushing the audio hardware of said consoles beyond what the creators of the hardware ever envisioned, sometimes with certain restrictions (if you give up on any kind of graphics or gameplay then what you can do with the NES is quite impressive, make yourself have some kind of gameplay or graphics and it is harder).
and Amiga and co sequenced music is more your thing then you can do it too. The N64 is an odd one -- it had plenty of great midi music (the devs behind banjo kazooie did a let's play when they were doing a crowdfunding session and what they were discussing on sound was fascinating) but at the same time was new enough that concert length CD quality wave type audio was no big thing and its sequencer side of things was not as... iconic as the 8 and 16 bit stuff that preceded it and the chips that supported it, or as charming as the C64 or some of the apple stuff. None of this stops you doing you though.
Had you wanted to do it on the consoles mentioned earlier I might have been able to link you some stuff and sent you on your way, the N64 however was a bit of a flop, is a pain to emulate, kind of hard to code for and never really gained too much of a hacking or homebrew scene. That said you are apparently not without optionshttp://www.neoflash.com/forum/index.php/topic,6311.0.htmlhttp://www.pouet.net/prod.php?which=51604
That leaves homebrew flash carts as I assume you are not going to want to sponsor $100+ flash carts for your run. Again the N64 is not my domain and a quick search for something like the homebrew flash carts for the GB/GBC and GBA I linked earlier came up short. I can see there being a path for you to buy a bunch of cheap sports games (be aware that despite the N64 being a flop and ageing horribly it is enjoying something of a resurgence and thus everything N64 is rather expensive at this point in time) and have some fun with a soldering iron but this post will not be the thing that links up a guide to you.