Life was.... not so hot back then.
Oh! I love your question...
To start off with, yes, Martha could have been the name of a muggle girl in 550 AD. But it is unlikely... I'll explain what I mean later.
No, the Statute of Secrecy had not yet been imposed... that came more then a thousand years later in 1692. Hogwarts is also still over 500 years away.
What was it like to live in Britain in 550 AD? Well....
This is all going to sound a bit grim, but here goes: banish all ideas of luxury, cleanliness as we know it, and leisure. Imagine a world where your best tools might be a plow, a rickety cart, a spade, an ox or two... there is definitely no time for primping in front of a mirror or sitting under a tree eating apples. Men and women must work from sunrise to sunset to scrape together a living.
Death and disease are daily companions. In the 550's Britain was struggling against plague in almost every city and major town. War was being fought everywhere, since Britain had not yet consolidated into one kingdom. One lord fought another bitterly over mere scraps of land. I'd say at least two thirds of all children would be expected to fend for themselves after the age of 13, either as apprentices, wives, or laborers.
Here is an example of what was historically going on in Britain between 550-560:
549 - "Yellow" Plague hits British territories, causing many deaths, including King Maelgwn of Gwynedd. Ireland also affected. Saxons, for whatever reason, are unaffected by it.
c.550 - Death of St. Ninian, Bishop of Whithorn. Birth of St. Tremeur. Murder of his mother, Triphine, by his father, King Cono-Mark of Cerniw, Poher and Domnonée. Prince Judwal of Domnonée retakes his throne. Cono-Mark flees to Cornwall. The semi-legendary Kingdom of Lyonesse possibly inundated by the sea.
552 - King Cynric of Wessex lays siege to the British at Old Sarum and put them to flight.
555 - St. Cybi Felyn, Abbot of Holyhead, dies at his monastery. Murder of St. Tremeur. Death of his father, King Cono-Mark of Cerniw and Poher.
c.555 - Death of King Erb of Gwent. The kingdom is divided into Gwent and Ergyng.
556 - King Cynric of Wessex lays siege to the British at Barbury Castle and is victorious.
558 - Broërec is attacked by King Childebert of the Franks. King Canao II leads resistance.
c.560 - Prince Elidyr of Strathclyde invades Gwynedd in right of his wife. He tries to expel his brother-in-law, King Rhun Hir of Gwynedd, at the Battle of the Cadnant Brook, but is killed in the process.
[Hope that made my answer to the name question plain. If your girl was Jewish, or lived in Christian Rome, or lived in a Monastery were they spoke Latin, her name could be Martha. If she was British born and named... you will have to start searching for names like the ones you see above (except those are all guy's names) Several girl's names from that era: Aderyn (bird), Almedha (shapely), Aneira (truly golden), Caoimhe (gentleness), Angwen (beautiful), Awsta (revered), Berwyne (fair-haird) , Bethen (their form of Elizabeth) Ceridwyne (fairy poetry), Eiddwen (beloved), Delyth (pretty), Heledd (traditional), Mair (bitter), Morwenna (maiden), Shyla (strong), etc..... I could keep going if you like.]
These people are tough. They have to be. Life expectancy would have been around 25-35 years. Anything above that would have been considered old. There was a very high child mortality rate: only 1 in 6 would reach adulthood. Hygiene doesn't exist-- they actually thought it was dangerous to wash the body, they deliberately put grease in their hair to make it 'shiny', they threw the contents of their bedpans out into the street or stable yard. It would not be uncommon to come across someone relieving himself against the wall of your house.
Witches and wizards would have been frowned upon by the muggle community, but secretly they would have had a well-beaten trail to their doors. In the classic book Lorna Doone, there is "Old Mother Meldrem" who gives the farmers daughters charms, fortune telling, blessings, etc. Everyone is terrified of her, but go to her when they need something badly enough.
That is sort of how I see wizards in 550s. They are scattered across the face of England. Some have managed to find several more of their kind and they settle together in a muggle village. But they must meet only by chance. Here and there a muggleborn is born, grows up, marries another muggle and spends her entire life selling clever medicines and poultices. Then there might be a child of two magical parents who has the advantage of both their knowledge. Perhaps he learns how to employ his skills more... skillfully.
This era, as far as I see it, was the slow self-recognition of the magical race. Each wizard and witch has only a very vague idea how many magical folk their are. I would guess very few wizards had actual wands made by wandmakers. Some would probably use whatever beth channeled their magic. This sort of relates to the stuff Olivander says in DH.
Wizards and witches mingle with the muggles, both using and hiding their powers as the see fit. Occasionally they would hear about the exploits of wizards like Merlin and marvel at what he did, probably not thinking of themselves in the same class. The real problem is that they have not begun systematically trying to learn their powers. Once wizards start banding together and 'snow-balling' the Magical world will really get up off the ground. What good is magic if you don't know how to use it?
But life for a young muggle girl will look something like this: She is strong, brave, well-endowed with common sense. She is physically fit, and knows stuff about animals that would make us pass out (like how to butcher them). She has probably helped her mother give birth to her younger siblings. She can cook a meal to fill a family, and can spend an entire day on her knees in a freezing creek washing clothes. She spends no time on her personal appearance. If she is blessed with looks they better be the kind that lend themselves to wild and untamed. She is not prim or prudish.
Your girl would wear heavy woolen clothing most of the year. Skirts to mid calf, bodice laced up, long sleeves rolled up or down. Underneath she'd wear a white skiff or undershirt. She might wrap her hands in strips of cloth to keep them warm. A shawl wrapped around the head or a hooded cloak for outer wear. Think bulky.
In summer she would have lighter clothing on... probably a belted tunic over her shift. It would still be rough spun cloth, unless she was very wealthy-- speaking of which, if she is wealthy all these points are true, but to a lesser degree.
I stress that life was dirty and tough back then. You only ate what you dug out of the ground, or else you forced others to get it for you. Magic would be envied because it makes things like food, shelter, and health that much easier to possess.
In all honesty you have my admiration for tackling this era of wizarding history. Much too daunting for poor little old me. I hope my ramblings were useful. I can be much more specific if you need me to.
Best of luck, and PM me anytime for source recommendations, thoughts, etc.
Moderator's Edit: Excellent post! 10 points awarded.