If mini's are ever grafted, like other roses, then it's just reverted back to it's original rootstock. That's exactly what it sounds like happened. It's a very common problem in frost zones (with larger roses) so most gardeners have opted to have at least some own-root roses in their garden. And, most have a Doctor Huey rose that has sprung up unexpectedly where another was planted.

In larger roses it would have reverted back to (red with yellow stamins) Dr. Huey. So, a good question to ask on here might be 'what rootstock is used with miniatures?'.
If you go to the JP website, or whatever other growers are known for propogation, the answer will be there.

To be sure, dig down around the rose bush a few inches. If the new growth has begun below a bulge in the stock, or below where the other branches originally were, then you've probablya got your answer.