Invest any savings into an ISA to avoid income tax and capital gains tax. I would assume that the vast majority of minimum wage earners won't be putting their money in to ISAs where the money effectively devalues currently because of the rate of inflation vs savings interest. Counterintuitive.
Invest in a stakeholder pension, so that they can put £2,800 in, and HMRC top it up with a further £700 so the actual investment is £3,500. A wonderful idea, but I certainly don't have the additional to spare, even though the government will top it up. I have my minimum pension contribution deducted from my wages - it's not for tax avoidance as I don't earn enough to pay tax.
If their earnings are really low, they could voluntarily pay NIC to guarantee a state pension. Am I missing a bit where somebody struggling on min wage has enough spare to make additional contributions??
They can reduce their working hours/earnings to qualify for higher tax credits, free prescriptions, etc. Fantastic idea, we should encourage all low paid workers to reduce their earnings as much as possible so they can claim more benefits 
The lower earner could transfer some of their unused personal allowance to their higher earning spouse. a f/t min wage earner uses their full personal allowance.
I am not against any individual minimising their tax liability via legal means, BUT as I have said before it is hypocritical to vocally shame the "wealthy" for doing so whilst simultaneously doing so yourself. If my understanding is correct, JC would have had to donate around £25k to charity to receive the £6k tax break declared on his tax return - he donated more via the donation and tax break to his chosen charity than a great number of his grassroots supporters would earn in a year to support their families. And that tax money does not go in to the public purse to pay for the NHS etc.
Imagine if Labour do win the GE and implement their new tax regime. Now imagine if all of those new additional rate taxpayers follow a similar legal tax minimisation strategy - will we call out the Labour government for effectively putting less money in to public services, or will we just support them in taxing everybody even more to make up the shortfall? Let's also not forget that millions of low paid earners are among the most generous charity donors; those who have little tend to give to help those who have less IME, albeit in smaller but frequent amounts - they can Gift Aid their donation but cannot mitigate their own tax liability.