I bought a '07 [ close out ] Honda 500 Rincon 2 months ago from House of Honda in St Cloud. Paid $6400 out the door. Single cylinder air cooled, mucho torque. I winter in Breezy Point and it gets the job done. I like it. I don't know if they have any remaining 07's. You might want to consider a decent condition used machine if it's your 1st.
I know Mies Outland in Watkins, MN had some new non-current Polaris 500 Sportsmans they were blowing out at less than $5000 a month or so ago. That is worth a shot.
Most of the wheelers I see on the Ice are either Polaris or Honda.......Match the machine to what you want to use it for and remember engine size equals weight.
The thing about cheap is you get what you pay for. At 4 to 6 thousand dollars, spend a little extra cash and get a 4 wheeler that will suit your needs and last. Quite a few dealers are discounting slow moving and last year models.
On a personal note: I drive a 1986 Honda 350 that is held together with duct tape and bailing wire, but it gets me where I want to go and is reliable. Old Red serves my purpose, I only use it for Ice Fishing. I have to admit, I do get some strange looks and a few chuckles when I pull up next to a new machine.
Remeber that some are harder to maintain than others. I have a kawasaki and I can get filters, etc at Napa or a parts store close by. It is easy to change the oil and keep up. My brother has the Honda and it is a bit of a pain to maintain. It is a messy one to change the oil on because of the type of filter it takes (not trying to insult Honda owners!). Something to think about.