Depends on the issue, but generally a new system is the most expensive option (with some exceptions, but power issues are typically not among them).
My PSP had power issues recently. A new battery for it was $20, and that was the more expensive fix. There was also some water damage around the fuses that could have been the cause. I had also turned it in for sound issues, also caused by damage to fuses from water. $45 for the fuse fixes, and I already said the price for the battery (I could have replaced the battery but it was already in the shop).
I'm not sure what a 3DS battery costs, but probably less than the new system and even some used ones. But take it to a local tech shop and get them to determine the issue. Usually shops will give a free diagnosis and estimate, especially if you stress you want to know the price beforehand.