Here is a link to the RV parks in/near Eugene:
Eugene, OR Campground Reviews - Best of Eugene Camping - RV Park Reviews
My first choice would be Deerwood and second would be Premier. I've never stayed at either one but a friend has stayed at Premier a few times. It is in Coburg but only a few miles from Eugene and it is right next to I-5 so if you don't like traffic noise, this isn't the place for you.
Deerwood is on the very south side of town at the junction of I-5 and Hwy 58 so there is probably highway noise there too.
Both parks are pricey. You will probably have to call to find out if they have any monthly spots available. Rent is very high here in Oregon and lots of people have purchased RV's and are living in RV parks now.
The other option might be a mobile home park. Some of them allow RV's to occupy vacant spots and the monthly cost might be about the same. Unfortunately, I'm in Salem so I don't know anything about the mobile home parks in Eugene.