This is a tricky question, as it centres around the end of Roman Britain (where the literary sources can be counted on one hand and the archaeological sources are pretty hotly debated), but I think I can unpick some of where this is coming from for you. One place this may be coming from is the idea that the Welsh are the descendants of the original Romano-Britons and have the "most" continuity of culture - as opposed to the east of Britain, where significant cultural change occured through the Anglo-Saxon settlement - and the other is pure mythology.

The "Romanness" of Wales is preserved in texts like the Mabinogion, but to what degree it was ever fully culturally Romanised is debatable. As far as we know it had relatively few civic centres, all of which are concentrated in the south, as well as several military bases and auxiliary forts which weren't abandoned until close to the end of Roman Britain (traditionally dated as 410, but Britain was certainly abandoned politically by 476 when the western empire from Rome collapsed) sometime in the 390s.

Some Welsh sites like Venta Silurum (Caergwent) continued to be occupied through to the early 400s before being largely abandoned, and other small civic centres essentially disappear. It's theorised that the Kingdom of Gwent took its name from Venta Silurum, which points to some continuity of memory if not necessarily language (as Venta Silurum essentially means "The Silures' marketplace"), and there are some post-Roman burials inside and outside the town, suggesting a population in the area - but not an urbanised one. Urbanisation essentially disappears entirely in Wales during the early medieval period. Other early medieval Welsh kingdoms seem to have formed with these abandoned civic centres at their centre, but it's unclear if that is due to them being the centre of any sort of Roman life or because they were deliberately originally established inside the area of influence of pre-Roman tribal institutions.

There is some evidence for continued contact with the mediterranean and France in the presence of imported pottery and glass at some high-status sites during the 5th-7th centuries, which are similar to high-status sites found in Ireland and western Britain, and other finds suggest that the coast at least was part of the Viking Irish Sea trading network, including Ogham inscribed stones in the kingdom of Dyfed. There's also evidence for monastic foundations during the 6th century, which are probably (at least initially) Insular Christian.

So on the archaeological level we can say that there is a great deal of disruption, as urbanised Roman sites are abandoned and Wales fractures into a number of petty kingdoms. This is extremely similar to the pattern which we see in the rest of the province; the main difference is that the centre of cultural gravity tips towards Ireland, Britanny, Cornwall and Cumbria as the east of Britain begins to be dominated by the English.

After that, we move into the realm of mythology and extremely shaky ancestry. Let's talk about the Dream of Macsen Wledig, which was a piece of orally transmitted mythology that was written down in the 12th-13th century. To summarise the dream: Macsen Wledig, the Emperor of Rome, has some dreams about an Extremely Beautiful woman named Elen. Elen turns out to live in Britain. He visits Britain, marries her and gives her father power over all of Britain and orders three castles built for her. Then! Alas! While Macsen is off fannying around with his new wife he is Usurped. With the help of her brothers, he takes Rome back, and then grants them Britanny, where they cut out all the women's tongues so that their children will only grow up speaking their father's language. It's a lot. A variation on this is also put forward by Geoffrey of Monmouth in the pseudo-historical History of the Kings of Britain.

So who is Macsen Wledig, exactly, and did any of this happen? He's traditionally associated with the Emperor Magnus Maximus (325-388), who pops up an awful lot in Welsh legend. His bid for power - taking a lot of troops out of Britain - is possibly the last gasp of Roman military presence in Wales, and if we're to believe Gildas he also took some of the people appointed to civic positions with him. It's possible, even likely, that he did appoint people to keep an eye on the province in the meantime, and there are repeated mythological claims that he married a British woman. As a result, many Welsh kingdoms claim descent from him or from his children, including a claim on an 8th century pillar that his daughter Elise (otherwise unattested to) married Vortigern, who may or may not (depending on your interpretation of Gildas) have been a high-ranking official at the end of Roman Britain. After this, we fall into a pit of Arthuriana that I am frankly not qualified to comment on. This - and folk songs like Yma o Hyd, which taps into a resurgent vein of Welsh nationalism - is most likely where your friend's claim that Wales was a Roman "rump state" came from.

In conclusion: Wales probably saw some continuity from Roman rule, but we don't know what that looks like, and the way in which society changed is similar to the rest of Britain. Many Welsh kings traced their ancestry back to Magnus Maximus, who did exist, but the line of transmission is extremely shaky and it's unclear whether it represents an actual blood tie or not.

The most credible, quoted “academic” quote about Wales being a “Roman rump state” would be by Bryan Ward-Perkins in “The Fall of Rome” (2005) p49 “north Wales can lay claim to being the very last part of the Roman empire to fall to the barbarians- when it fell to the English under Edward I in 1282.”

He is echoed by Dr Rhun Emlyn, (medieval historian at Aberystwyth University)

“An interesting feature when considering Wales in this period is that Wales, before it was conquered by Edward I in the 1280s, was the last remaining part of the Roman Empire in western Europe that hadn’t been conquered by ‘barbarian’ peoples.”

I would suggest that this is separate from the (clearly false IMHO) identification of the mythic Macsen Wledig with the very real co-Emperor Magnus Maximus who as a result appears in genealogies of Welsh royalty, and more a combination of the Kingdom of Gwynedd roughly following on as it was left by the Romans, only under the rulership of a dynasty claiming decent from another Roman appointed commander/Vassal king.

Maximus was declared emperor in north Wales, almost certainly leaving some sort of authority in his name as and went to press his claim in Europe. He also went north between the walls (Hadrian's and the Antoine) to the territory that the “Imperator” Cunedda is supposed to have come from.

While not all historians believe "Cunedda" was his actual name, and some believe he didn’t even exist many do, or that he at least represents a real migration of the Votadini from the area around Stirling to Gwynedd and Anglsea. Even if he doesn’t, it’s a telling point that the Welsh ruling dynasty took part of their right to rule from Roman authority.

Cunedda’s grandfather Padarn has the nickname “the red tunic”, which cause many to think that he was either a tribal leader of a buffer state who had been given an honorary Roman commission or a Roman solider that was given command of one of 4 buffer states created in between the walls, which may at some point have been the unlocated British province of Valentia.

I’ve also heard the theory that Padarn’s father Tacticus was Crispus, oldest son and heir of Constantine the Great until he was executed in 326 and subjected to damnatio memoriae. It seems outrageous, but Constantine the Great was born in York, a minority of historians think his mother was a native British Royal, and there were a number of unusual visits to the buffer states by emperors, and senior officials strengthening the boarder with troops but giving unusual levels of autonomy.

The native Gwynedd dynasty passed through at least one female line, but presented themselves as having been the same dynasty ruling Gwynedd since Cunedda, and even traced themselves back through a female ancestor of Cunedda to the legendary Welsh King Eudaf Hen.

It may be worth pointing out that each of these female line switches would have disqualified the family from continued rule under “Welsh law” and in the other Welsh kingdoms. Not only did in not do so in the Kingdom of Gwynedd, but their kings also took control of other Welsh kingdoms through force and the “reasoning” that their wives or mothers were heirs even though there were male heirs of the old dynasty. Obviously Roman emperors sometimes adopted others as heirs and son-in-laws sometimes succeeded to rule the empire.

This “Roman” way of doing things (allowing decent through female lines) was really only evident in Gwynedd and in the Pict region which boarded with the Votadini/Cunedda’s territory for the longest time.

Also interesting is the Welsh Red Dragon. Cunedda’s great-grandson was called “the Dragon of the Isle”. The Roman cavalry cohorts used Draco’s as standards sometimes. Also the shield markings of a Legion based in north Wales had what looks like two serpents or Dracos on them. It doesn’t prove that the Kingdom of Gwynedd was “Roman” but when added to their creation story it gives something to ponder.