How old are you?
Assuming your mother has health insurance--if you're a student and under a certain age (the age cutoff may vary by state), you could be added as a dependent under your mother's US health insurance. Add to that a high deductible and your status as being intermittently present in the US, and you might get a better rate as a dependent on your mother's insurance than on your own.
Travel insurance to cover you for going home to your home country could potentially be dicey. If something seriously goes wrong that lasts several days or weeks that could financially wipe you out (ex. long-term intensive care stay or surgery), travel insurance is primarily designed to get you home. If you're listing the UK as your "healthcare insurance home" for your travel insurance, the coverage may therefore entail having you brought back to the UK if you require the kind of long-term care that would make it impossible for your to continue your studies and keep your visa.
In short, if something catastrophic happened to you, you had to leave school, and had to stay in the hospital/long term acute care for months, where would you want to be? Your answer to that question is what travel insurance will help you get to. As I don't know the answer to that question for you, I cannot advise you as to whether or not you should obtain US or UK based travel insurance.
It may be more financially viable for you to forego insurance during your stays in the US while you're a student...or it may not. Other than being a nice thing for all people to have, can you tell us a little more about why it's important in your case to have health insurance during your visits in the US? Do you have pre-existing conditions, either active or in remission? Are you receiving ongoing treatment for any conditions? Or are you looking for coverage for emergencies? Is healthcare access your (mother's) concern, or paying the bill at the end?