Moving the website to another host may not be so simple. Like Anatoray mentioned, if all the contents were pure html, it would be easy. However, it becomes more involved it her website involves dynamic content and backend (database) access, where you would have to move data also.
Again, updating the website depends on what technology was used to begin with. A lot of these web design companies use a set template and just make your website through their own content management system, in which case, it would be rather difficult to do. Companies do this so you would eventually pay them more to do maintenance, as is in your mother's case. However, if your mother's website is pure html and all static content, then you just have to learn html to update it once you find and move to the new host.
Cost to update a website really depends, like I mentioned above, on what technology was used to begin with. I am guessing it would range from $20 (simple html) to $150 (heavy coding + backend stuff) per hour, depending on how involved it would be.
It really can't be determined until you find out how was your mother's website was constructed and how often and involved her future updates would be. Hope this sheds a little bit of light on your question.
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