A few things.
The very first iPhone Sprint got was the 4s and by extension the 4.
There has never been an iPhone 3. There was the iPhone 3G (because it had 3G) but that was AT&T only at the time.
The fact that the phone has no SIM slot tells me that it's an iPhone 4. The Sprint iPhone 4 is the ONLY Sprint iPhone model that does not have a SIM slot - therefore if it has no SIM slot, and it's a Sprint model it is an iPhone 4. It uses an internal SIM-like circuitry and that circuitry is locked to Sprint.
The route you are taking (jailbreak, unlock) implies that you have read about and are seeking a software unlock. Well, the software unlock days died about four or so years ago. With the advent of $1 unlocks and free unlocking by the carriers development ceased.
You say the phone is on 7.1.2. So, even if you could still find something (a software unlock) the version of iOS is too new to use it.
In short, there is no way to do what you want. Even if you could "unlock" a Sprint iPhone 4 you have no capability of inserting a Straight Talk SIM.
Also, you may have heard about MSL codes at some point. Forget it. Apple does not use MSL codes to "reprogram" iPhones. They are less than useless.
I'm sorry, but unless your cousin's mom is willing to use Sprint this phone is not going to any other network.