Hi,
A family member has a 3 month old Great Dane/Lab mix that needs to be re-homed. ... I'm thinking that we're not ready for a puppy... but she's so cute!
Not meaning to burst your bubble here, but my initial thought was...."we're not ready for a puppy"...then you shouldn't have one. Puppies are always cute, so don't let that be the biggest part of this lifelong decision. See if you can find a short-term foster, until you are ready. Maybe if you have another month to consider housing options, get supplies, puppy-proof the home, etc, then you may be more ready. And when you're ready, you're ready. And you will know it. Don't be too hasty though, because pups that young certainly are a lot to handle.
Another side note...DO NOT send her to someone you do not know and trust. Make it clear that you will want her back within a month or so. And be sure to keep close contact and spend a lot of time visiting the pup until the time comes for you to take her home. This will allow for a smoother transition, as she will get to know you, and you can get to know her as well.