Thanks for writing. You'll need the same things you need for any mortgage refinance - credit, income and equity.
Since you have a legal lien with your relative, you're still just looking to pay off one loan with another, which is a refinance. On credit, most lenders want to see a credit score of 620 or higher before approving a refinance - and higher is better. If you have a low credit score, you may want to look into refinancing into an FHA loan, which often has better terms for borrowers with less than perfect credit.
On income, your total monthly debt payments after refinancing, including the new loan, should not exceed 43 percent of your monthly income. Some lenders will go higher with good credit.
On home equity, you can generally refinance if you have at least 5-10 percent equity in the home. You'll pay a higher rate for a second mortgage if you have less than 20 percent equity, though.
You also might consider a cash-out refinance that of your primary mortgage that will allow you to consolidate your primary and second mortgages into a single loan. You may be able to get a better rate this way, but the closing costs will be higher than refinancing the second mortgage by itself.