Ouch. What make and model phone is it? Just that there might be some comments online. Presume this is a PAYG phone not on a contract...
If you have an Asda and Asda Living then you may find bigger variety in Asda Living.
They seem to be one of the few places you can buy a PAYG phone without being forced to pay for a top-up as well.
Trying to get a PAYG in Carphone Warehouse, you'd need to also pay 10 (card) or 20 (cash) on top of cost of phone. Argos used to sell the phones without top-up but I think the networks must have threatened them and they now require a top-up fee too.
I know you don't want to lay out extra cash, but an unlocked one should be usable (don't know about Three network) with any of the UK SIMs.
Asda SIM works over Vodafone network and if they give you a goodwill credit then the cheapest unlocked (probably about 25 quid) could be used until that cash ran out.
Also, to give them some real "consumer" issues instead of bl**dy election topics, you could contact Radio 4 "You and Yours" and ask if others have as many problems with faulty mobiles from Vodafone.
You might find it worth complaint online to Vodafone, and tell them that you are about to ask the BBC if they have had reports of problems with Vodafone replacing phones. It will give them a final chance to send you a brand new phone.
If you key in * # 0 6 # (no spaces) the mobile phone should give you the IMEI (like a serial number) or that may be on the box.
(Just something to note so you can be sure whether they have sent you a NEW phone or your own faulty one back.)
Perhaps you can say to Vodafone that they have a week to get you a new phone... and then you could give Radio 4 as many details as possible, explaining that you liked the style of phone and facilities like the camera (to take photos of your DD) but that Vodafone has messed you about, and after the history of returns all you want from them is a new working model that will switch on and work, not for a day, not for a week, but for the next 12 months and longer.
If the story is taken up by "You and Yours" then I bet that as soon as one of their researchers makes contact with Vodafone, there will be a new phone on the way and an apology on the radio about how sorry they are that you had problems with your phone.