You'll have better luck asking this question over at /r/genealogy, but according to the Oxford Dictionary of Family Names in Britain and Ireland, "Bedwell" (also spelled "Bidwell" and "Bidewell") comes from the Old English word bydewelle, which is a "spring or stream in a shallow valley".

But likely not just any spring or stream in a shadow valley, because the books says that it is a locative surname that showed up in a handful of specific places in England that were home to a bydewelle. These gave rise to the places known as "Bedwell" in Essex and in Hertfordshire, to "Bedlar’s Green" also in Essex, to places called "Bidwell" in Bedfordshire, in Northamptonshire, in Devon, and in Somerset, and to "Biddles Farm" in Buckinghamshire. Some of these places are no longer known by those names, but they once were.

The book cites a "Stephen de Bedewell" on the Closer Rolls of Essex in 1229 as the first written instance of the surname. But as mentioned, there appear to be multiple origin points for the surname adopted by unrelated families.

For further information, you'll need to dive into the genealogical records. If you aren't in the U.K. (and even if you are), there's always a chance the surname was changed from something else, Anglicized from some other name. Proper genealogy is very historical, but it's also very specific to usually a small group of people, so unless an historian here has done a study specifically on the "Bedwell" family, you're probably not going to get much more from this sub. But the /r/genealogy sub has an active community and can be very helpful in getting you on your way of tracing your roots back. The U.K. has well-kept records with many of them digitized and searchable online, so even if there is a missing record or two along the way, there's a good chance you will be able to create a paper trail taking you back to the town in the U.K. your family name came from. That is, if the surname truly is English and wasn't Anglicized from some other surname.

As a start, I recommend you signing up for a free account on FamilySearch.org, which has a lot of free records that might help you get started. Also note that, in some cases, much of the genealogy has already been done, and all you have to do is go back a few generations and "plug in" to the family tree that's already well-researched. For instance, there's a book called A Bedwell family, Robert Bedwell, Sr. : fourteen generations of his descendants from St. Giles Parish, Cripple gate, London, England in 1635 to all the fifty United States of America and many foreign countries in 1982 by Larry King listed on FamilySearch. Your great-great...-grandparents may already be listed in the book and all you have to do is get the book, and trace back a few generations through census records to make the connection. There are other books, too, on FamilySearch related to the "Bedwell" family. After creating an account, click Search >> Catalog >> Search By Surname, and type in "Bedwell". From there, you'll see a list of books. You'll also want to search other spelling variations--quite often, even if not Anglicized, people changed the spelling over time. But note that not every genealogy book is well-researched. Even if the book cites a paper trail, you'll want to double-check it by tracking down the same records yourself. If the book cites no paper trail, it's best to take it with a grain of salt since many "genealogists" are amateurs who made assumptions that don't always pan out.

And also, unfortunately, some of these types of books can be difficult to track down, being only available in a select few genealogical libraries, or in the genealogy section of random public libraries. But they can often be worth tracking down--Interlibrary Loan is your friend when it comes to genealogical research.

Another place to help you get started is Worldcat which has a list of books on the "Bedwell family" as well.

More consistently researched will be the publications of the premiere genealogical and historical societies around the world. While I am not as familiar with those outside the U.S., in the U.S. for a start there is the New England Historical and Genealogical Register and American Ancestors published by the New England Historical and Genealogical Society, the New York Genealogical and Biographical Register published by the New York Genealogical and Biographical Society, the Virginia Magazine of History & Biography published by the Virginia Museum of History & Culture, the Pennsylvania Genealogical Magazine published by the Genealogical Society of Pennsylvania, and American Genealogist that covers the entire U.S., to name some of the older and more respectable ones. Each state and many counties will have similar publications. The ones I listed contain peer-reviewed genealogies (not every genealogy journal does that, but the good ones do), and so if you know what location your family came to/immigrated from, it's worth looking through the indexes of these journals to see if there's any articles written about the family. Again, like the books, much of the work may already be done for you, and all you have to do is "plug in" to the existing family tree instead of re-creating it from scratch.

Obviously, that's U.S.-centric, but many other countries and their provinces/counties have similar genealogical journals.

The document you linked to lists a handful of other American progenitors of the name aside from the Robert Bedwell Sr. whose family is the subject of the book mentioned above. These other immigrsnts might match up with some of the other books on Worldcat or FamilySearch, and I would suspect that, given the otherwise non-specifics in that document you linked to, those other progenitors have some research done on them. It's just a matter of finding it. It might be in some book, or in some old article in a genealogical journal.

With the Coat of Arms, I'd take it with a grain of salt. It's neat to look at, but those English heraldry Coat of Arms for specific families are mostly bullshit. Further, unless you've done the research, your roots are just as likely to be from any of the other places named in the Oxford Dictionary as Bedfordshire that that document and Coat of Arms mention. There seem to be multiple families called Bedwell all named for the Old English word bydewelle but after several completely different geographic locations. Despite the name, the families aren't related. So the Bedfordshire "Coat of Arms" wouldn't really apply to you if you trace your roots back to, say, the Essex line, or the Robert Bedwell Sr. of London line. But the Coat of Arms are kind of fun. If you find you don't trace back to Bedfordshire, you can even make up your own new Coat of Arms!

Anyway, that's all I've got. I hope this helps. Good luck!