I was actually spurred on to get my first cat (the beautiful Radar) because we had a horrific indoor mouse infestation, they were all over the place. So I don't think it's an unusual or bad reason to get a cat, as long as you're going to care for it properly in terms of food and vet care and give it some attention and affection.
In terms of breed, (and I usually would not say this in breeders corner, but this is my advice for this particular situation) you don't need any particular breed if all you want is pest control + a warm furry creature to love and watch TV with of an evening. A young adult from a shelter would be ideal as they are in their prime of life and can be very predatory, it will also already be neutered. It will need to be wormed and de-flead regularly (every 3 months) if exposed to rodents, in addition to having the usual annual vaccinations and vet checks annually or more often if it becomes unwell. And good quality food - this does not stop them hunting, a well fed and cared for cat makes the best hunter, as they will hunt for fun if there is prey to be caught and they will be in tip top condition to do so.
Now in terms of specific breeds, my most prolific hunter is part Cornish Rex. I don't think there is anything about that breed in particular that makes them great at hunting, other than they are a very athletic and active breed of cat. If you want a particular breed, then I would suggest looking at athletic slender breeds of cat as they tend to be fast, alert, and very active.
If you want a great hunter but also a very loving lap cat that will want to be be there with everything you do as part of the deal, then Oriental, Siamese, Abyssinian, Sphynx, Cornish Rex, and Devon Rex are great cats in terms of being very active at the same time as being absolutely devoted to their human family. Bengals and Ocicats are also very active cats but tend to be a bit more independent and less needy - they still love to interact with you but don't feel the need to be velcroed on to you at every moment. Burmese are also active cats. IMO though all these breeds can be quite high maintenance (very demanding of your time and attention) in various ways and if you decide on particular breeds this is something that needs to be considered.
But realistically, cats are among the most predatory animals on the planet and most cats, regardless of breed, will likely sort out or at least make a huge dent in your 'huntables' problem, getting cats certainly sorted out my rodent problem - but it's the personalities of certain breeds that makes me prefer Orientals and Cornish Rex - because of the superb companionship they give