Your question isn't great, but I will attempt to answer this piece as it seems really the root of your personal finance question:
I want to convince my wife to make this move because it will save us
at least 800 month, but she fails to see how buying a second home is
financially sound because we have to lose our savings and we have to
pay interest on our second home.
Her logic is it will take almost 5 years to get back our down payment
and we have to pay interest as well. So how can this move help our
family financially in the long run? ...
Is she right?
She is mostly wrong. First, consider that there is no "ROI" really on your down payment. Assuming you are paying what your home would sell for the next day, then your "RIO" is already yours (minus realtor fees). She is talking about cash on hand, not ROI. I will use an example without taking into account risk of home markets going down or other risks to ownership.
Example: Let's say you pay $2800 a month in mortgage interest+principle at 5.5% apr and $200 a month in taxes+insurance on a $360k loan ($400k house). In this example let's say the same house if you were to rent it is $3800 a month. Understand the Opportunity Cost of renting (the marginal amount it costs you to NOT buy). So far, your opportunity cost is $800 a month.
The principle of your house will be increasing with each payment. In our example, it's about $400 for the first payment, and will increase with each payment made while decreasing the interest payment (Suggest you look at an amortization table for your specific mortgage example). So, you're real number is now $1200 a month opportunity cost. Consider also the fact that the $400 a month is sitting in a savings account of sorts. While most savings accounts give you less than 1% in returns and then charge taxes on that gain, your home may (or may not be) much higher than that and won't charge you taxes on the gains when you sell it (If you live in it for a period of time as defined by the IRS.) Let's assume a conservative long term appreciation rate of 3%. That's $12k a year on a $400k house. So, now you're at $2200 a month opportunity cost.
In this example I didn't touch on your tax savings of ownership. I also didn't touch on the maintenance cost of ownership or the maintenance cost of renting (your deposit + other fees) which all should be considered. You may have other costs involved in renting. For instance: The cost of not being able to fully utilize your rental as your own house.
This may be an even simpler and more convincing way to explain it:
On the $2800 mortgage example, you will be paying around $19k in interest and $2400 on taxes, insurance = $23k per year (number could be way different in your example). That is basically throw away money you're never getting back. On the rental, 100% of your rent at $3800 a month is throw away money you're never getting back. That's $45,600 a year.