I hate to answer a question with questions, Kristina, but:
1. How "little"?
2. How "old"?
3. How "luxurious"?
There are many of the touted 10,000 lakes of Minnesota which fit the definition of "little old cabin on a lake". Luxury, though, is often in the eye of the beholder. For some, luxury is 300 ft of frontage and plenty of privacy. For others, luxury is a 5 BR / 5000sqft palace on Gull Lake.
If you're looking for yourself or for a client, my recommendation is to tap the Realtors to the North - they know the territory; they have the listings. Take some time getting familiar with a few different areas and don't forget accessibility as a selling point. Anyone who has been in the logjam called I-35 South from Duluth on a Sunday night in the summer can attest to the "closer might be better" philosophy.
There are also many communities within an hour or two drive of the Twin Cities which are truly the best of both worlds - a home on a lake and a decent commute to work. You leave the city behind each weekend and "go home to your cabin"!