There is (virtually) no difference between "zero interest" and "0.01% interest" from the IRS's point of view. The relative must charge you a reasonable interest rate (see the Applicable Federal Rate as Nathan L pointed out) or the difference in that rate will be considered a gift; so 0.01% and 0.00% are different only in that 0.01% less is gifted.
Assuming this is a "long term loan" (over 9 years), the current rate is 2.75%; so for $100,000 that would mean annual gift of $2,750. Because the current IRS gift tax exclusion is $14,000, you're well under that unless they give you other gifts. In fact, if they choose to make the loan itself a gift, they could simply forgive $11,000 worth of loan payments each year (eventually $12,000) and be still under the gift exclusion, effectively making this a tax-free gift entirely (over the course of years). (Of course, the IRS may decide to treat this differently if you never make payments; talk to a CPA if you're planning to go this route.)
Second, though, is the issue of the income for the lender. If this is over $100,000, the lender must treat the interest as income, even if not charged ("gifted"). So that $2,750 (to begin with, obviously it goes down) is income for the gift-giver each year; they should consider that in how they choose to implement this loan.
As such, you might want to consider paying interest equal to their income tax on the interest; if they are a marginal 28% payer for example, then 0.28 * 0.0275 means you should pay around 0.75% interest (or $750 initially) to cover their income tax. I would consider this appropriate if I were making the loan.
You can also structure the loan differently to avoid some of the interest, by the way. Not only can they choose to gift you $14,000 pear year for some period of time, but they could also structure the payments such that you had a "balloon payment" at the end of a nine year repayment period. Perhaps by the end of this you'll have a much better job and be able to pay back everything by then. If so, you could get by with the lower 1.97% interest rate, meaning you'd have to pay a bit over 0.5% interest to cover their taxes. (This would be something like, for years 1-9 pay 0.5% of the principal per month, which would be $500 per month and would pay 6% of the principal per year ($550 with the interest); then at the end of year 9 the other 55% of the loan becomes due immediately. You can of course increase your payments over time, and can work things out to make this work for you, but the point is if it's 9 years and done you get to "charge" 1.97% while 10+ years it has to be 2.75%.)
You also could avoid the interest-income issue if the loan amount is under $100,000; meaning after the first year or so it may be moot. See IRS publication 550 for more details, specifically:
Limit on forgone interest for gift loans of $100,000 or less. For gift loans between individuals, if the outstanding loans between the lender and borrower total $100,000 or less, the forgone interest to be included in income by the lender and deducted by the borrower is limited to the amount of the borrower's net investment income for the year. If the borrower's net investment income is $1,000 or less, it is treated as zero. This limit does not apply to a loan if the avoidance of federal tax is one of the main purposes of the interest arrangement.
Or the family member could loan you $99,999 and then gift you the extra $5k or whatever is over that, of course, if it's close.
See this Marketwatch article for more details.