My British mother wants to stay with my sister in Florida for about a year. How can she do this? She doesn't want to become a resident and has a home in the UK. She has been over to the US with an ESTA but they only last 3 months.
Based on the question and comments here are the two options that are fitting for the case:
Option A) If the intent is just to visit for 1 year she must apply for a B1/B2 visa in order to stay for that period of time. It is the Border Officer discretion to determine how long you can stay. Generally, they stamp for 6 months. More or less time may be given according to the circumstances when you are at the Point of Entry. It is always possible to leave the US (to Canada or Mexico for example) and come back again in order to get more months.
Option B) If the intent is to PERMANENTLY STAY in the US (i.e. work, live and pay taxes here) then she must apply for a green card. She's a Immediate Relative of a USC, therefore, eligible for a green card right way. Your sister will apply for a i-130 (petition for alien relative) and, once approved, your mother will apply for a IR-5 visa in the UK. The whole process usually takes around 9 months. After 4 years and 9 months your mother will be eligible to apply for US citizenship.
It is important to note that if she becomes a green card holder she cannot leave the US with the intent to establish residency in another country. If so, she will loose her green card. Only after becoming a USC that she would be able to return to the UK to live there (there are exemptions to this rule).
In regards to taxes for either option, that's a different story with several different factors.