Before getting into how you can do to boost your credit score, however, I want you to be aware that a great credit score alone is not enough to get approved for a mortgage loan. If you don't have sufficient income, no matter what your credit score, the banks won't even bother looking at your mortgage application.
Of course, what is sufficient income will depend on the cost of the house you're looking to purchase. And, indeed, it also depends on your credit score, because with a better credit score, you'll pay a lower interest rate, and that has an effect on the monthly payments as well.
When looking at mortgage applications, in addition to credit scores, banks primarily consider your debt-to-income ratio which is how much debt you have compared to your overall income.
Generally speaking, your debt-to-income ratio after taking out the mortgage should be below 36 percent, preferably lower. This means that the sum of all of your monthly debt payments, including your mortgage (principal, interest, taxes and insurance) as well as student loan payments, car loan payments and credit card debt payments (which fortunately you don't have) must not exceed more than 36 percent of your monthly income. For loans from the Federal Housing Authority (FHA), the debt-to-income ratio required is even lower, at 31 percent of your income. If you are looking for a loan with a small down payment an FHA loan for first-time home buyers is your best bet as the 31 percent debt-to-income ratio will apply to your situation.
To find out your debt-to-income ratio, try this handy calculator from Bankrate.com. The Federal Housing Authority also has several convenient calculators to let you estimate how much you can borrow. This will help you greatly when it comes time to look for a house.
Another thing to consider in addition to improving your credit score is to build up an emergency savings fund. The fact that you fell behind on car loan payments while unemployed last year should be a red flag to you. You could get laid off from your job again, and, if you then fall behind on mortgage payments, you could lose your house. You don't want your dream of owning your own house to end in the nightmare of foreclosure.
To protect yourself, the ideal is to build up an emergency savings fund with at least six months of living expenses. If you are not able to do that before buying the house, you're better off buying a less expensive home so you'll have enough money left over to continue building up your emergency savings after you buy. It's better to have a less expensive house with enough financial leeway and peace of mind.
OK, after this lengthy preamble, let's take a look at how you can boost your credit score. From the details you gave, it sounds like there are two factors weighing down your score: One is the late payments on your car loan, the other is the fact that you don't use credit that actively. Even though you have one credit card, if you don't use it, it won't help your score as much as it should.
Fortunately, there is a simple solution that will help address both issues: use credit more actively. Begin by using the credit card you have, charging several items to the card each month. Follow the basics of good credit card management: pay bills on time, don't carry more than 10 percent of the card limit over from month to month and preferably pay the balance off in full each month.
To further improve your credit score, apply for one or two more credit cards, and use them as well. You can alternate use between cards every three to four months if you don't want to keep track of multiple credit card payments each month. Applying for new credit cards will lower your score slightly at first, but the effect is temporary, and it will more than be outweighed by the positive impact of increased credit usage. Just make sure to open the new cards soon - well before you apply for a mortgage.
The marks on your report from the late car payments will continue to stay on your report, but once you introduce a more recent history of positive credit usage, their effect will become less and less. For the best mortgage terms, your score should be at least 720 and ideally over 740. But with your score already at 700, the good news is that you don't have that far to go.