What are the options to maintain a backup phone while travelling in the US?

You don't need a backup phone as a student in the US. This is for a few reasons:

• Most of the US is relatively safe. Especially as a student you are unlikely to find yourself in a survival situation where you need a second phone.
• Most of the US has decent cellular coverage, and multiple providers allow roaming on each others' networks. Many years ago you might have wanted two different phones to get better coverage in different regions, but this rarely applies anymore.
• USB "power banks" (external batteries) are cheap and widely used.
• Prepaid phones in the US are not nearly as common as post-paid, nor are they as common as they are elsewhere in the world. You won't find as many options available, they will be more expensive than they should be, and people will look at you funny. :)

First factor you need to consider is that the USA uses different GSM frequencies than the rest of the world, so would your older handset work in the USA? Global frequencies are 900 & 1800 whereas the USA & Canada use 850 & 1900.

There are a number of non-monthly pre-paid plans available, where you buy $XX and get YY months of validity. Many of these tend to hover around the $25 gets you 90 days, $100 gets you one year, with calls costing 10 cents a minute (both inbound and outbound, a sneaky US phone company trick).

Some of these pre-paid plans allow you to bring your own phone and simply get a SIM. Others require you to buy one of their phones. AT&T and T-Mobile offer bring your own phone plans, Tracfone has buy their phone (but they offer some really cheap talk only models).

Finding out the best plans from afar is not easy, as the phone companies tend to bury information about their cheapest offering hoping you will sign up for a more expensive monthly plan. I found talking directly with the sale people in their respective stores revealed more low cost options than their websites do.