It would help to have a bit of information on what your daughter plans to use the Cellphone for during her travels? Given the fact that you don't want to be locked into a data plan, that will probably preclude using T-Mobile as they would likely require at least a two year contract with a data plan. One other point to note is that I believe T-Mobile data service only includes 2G, and customers have to pay extra for 3G or 4G (where available).
Given the diversity in the places she will be visiting, you might consider using a plan with one of the travel phone firms such as Roam Simple, Cellular Abroad, Telestial, Mobal, Lebara, etc. (there are others but these are the ones that come to mind). Billing is often post-paid (calls charged to a credit card), so no need to top up. Check the respective websites for pricing and other details.
One option you could consider would be to use a basic unlocked quad-band GSM phone such as the Motorola V551 (check E-Bay or Amazon to see availability and prices). I used that model before switching to an iPhone and it always worked well. This would provide voice and text and would be a cheaper option than using a Smartphone, as data is typically quite expensive. They typically offer plans with a 30 day maximum limit, so you'd have to buy two terms for the Nigeria portion.
Especially in Nigeria, she could send you a text every day or two just to let you know all's well. For E-mail and other internet-based uses, pack along a tablet or a cheap Netbook for use in Wi-Fi areas (which will be readily available once she gets to Italy and Paris). A Netbook would be easier for handling photos, since many models have a built-in SD Card Reader. With a Tablet, a photo reader accessory usually must be purchased separately.
Regardless of which gadgets she decides to travel with, she will need to address the issue of voltage compatibility and Plug Adaptors. Will she be packing along any other electrical equipment?
Happy travels to your daughter!