Cheaper bike is not the solution. I rode on a bike worth 50 pounds for two years. Replaced the rear tyre after a puncture with a 15 quid replacement and it was stolen the very next day. From outside the British transport police station at Victoria
I don't know how it is in "nicer" climates, but where I am, my commuter bike has gradually accumulated various things:
- taillight which I don't bother to remove
- expensive headlight which I do remove when parked (but I leave the mounting bracket)
- pedal straps (a must, in my opinion, for safe riding in traffic; feet slipping off pedals can mean a crash)
- thorn resistant tires
- stronger wheel (I'm above-average height and not fat, but the cheapo wheels kept getting bent)
- fenders to reduce splashing in the rain
- rack to hold saddle bag
The bike itself is rather old and worth less than $100. All the accessories add up to maybe another $400. So, I do have a cheap bike, but it's still not so cheap to replace, and doing so is a rather big time commitment.
I mount head and tail lights on a helmet so I don't have to care about that part. You can also put lights on your bags too. The lights are often the most expensive part.
For stronger wheels & tires, I'd just get a thicker tire bike. For a strong frame I just go with cromoly. You don't need anything special then as far as strength goes. The rack, fenders and strap can be had for $50 total after all of that.