Sounds like a fun project. I know you said "simple," but here are my thoughts anyway.
The data you're writing the file over with isn't random, and one pass will still leave traces of the original data. It depends on the storage medium. For example, with magnetic devices, there's magnetic force microscopy. Even after ten rounds of the same thing, I'm not sure you're adding any extra benefits.
Renaming and deleting files won't add robust protection because, on most file systems, it simply changes the pointer to the location on the disk where the data is. Deleting a file only removes the pointer, so the disk sees the blocks as "available" even though the file is actually there. (This is why recovery software works.)
Thorough, secure deletes have the following features (software-based, not considering hardware solutions):
PRNG (Pseudorandom number generator) which generates random values to write at every byte of the file's allocated space (also see the encryption feature below).
Multiple passes (careful here, depends on the medium)
A run with all 0s then all 1s couldn't hurt, either. It may help interrupt a predictable, yet subtle, pattern of randomness (because computers aren't truly random).
Some file systems are "Copy-on-write", which is like a type of "revision control." These try to avoid overwriting data already in place. That protection would have to be circumvented.
RAID devices mirror changes to a disk onto another disk.
Fragmented files may start at one sector and finish at an entirely other part of the disk. Alternatively, anti-fragmentation features may keep redundant copies of the data or relocate them in real time.
Solid-state drives manage the disk space differently than magnetic drives. Coupled with wear-leveling, there are technical reasons that make secure wiping a little tricky. (See Wikipedia link below)
Encrypting a file before wiping it will help fill it with "random" bits... but really the best answer to secure shredding is to encrypt it before it ever hits the disk.
Don't miss this great question which talks about wiping info on solid-state drives, particularly flash drives. Remember that SSDs wear out after being written enough times.
Look at this article on Wikipedia for some more detailed background info.
By the way...
I'd really like to see a shredder that grants some plausible deniability. It would securely erase the file, then plop remnants of a decoy file in its place, perhaps a file chosen by the user that exists elsewhere on the hard drive. It would appear to be the remnants of a copy of that file which at one time was pointed to by the file system, was deleted by the user, and, depending on the size, was potentially being gradually overwritten by regular use.
If a forensics agent were to examine the portion of a disk where a regular shredder was run, it's easy to tell that it was wiped with random data. But if a decoy file was put in its place, I imagine it would be harder to tell.