I am of the opinion that your water heater is not the only reason you have a high electric bill, unless your family is taking many showers, doing many loads of clothes, and a lot of dishwashing. My main question for you is: Does the water heater have an insulating blanket, especially if your basement is an unheated space?

And as for that bid of $3100 for an install of a water heater...that's nuts. You are being ripped off. Get new bids. A decent water heater should be several hundred dollars, and tapping into the existing oil piping to the furnace to provide oil to a new water heater should not be very expensive.

I want to revise my comments about costs, as it's been several years since I replaced my own water heater and prices have gone up more than I thought. It's also unclear to me whether you are talking a regular oil-fired water heater or a tankless one. But in any case, I still think $3100 is much too high. I also am not convinced you will see big enough savings to justify whatever the outlay is. Your first priority should be to figure out why your electric bills are so high and what part of that is due to the existing water heater and its efficiency or inefficiency, how well it's insulated, and whether you can make changes in your hot water usage. To start with, contact your electric utility...many of them have energy conservation departments that can give you good advice, or maybe even visit your house to help figure this out. Your best bet might be to install an efficient, new electric water heater and try to limit hot water use more, and/or try to do things like running laundry loads late at night (off-peak hours) when electric rates could be lower.

Here is a link that might be useful: Oil fired water heaters