Does a widow retain her married last name after her spouse's death?

Question

I'm reading the books at the moment. I'm around the part where Lysa is ruling the Vale. It says something along the lines of

All the knights were trying to marry her so they could gain power, riches, etc.

It says that people marry into the houses. But when their spouse dies does that mean they're still in the house and able to have offspring in said house from other people?

ie. Could Lysa (Tully) Arryn matrilineally marry somebody and her children be of House Arryn? or would it revert and her offspring be of House Tully?

Answer 1

Widow's retaining names of their dead Lords

Other than what Mooz has answered, we have:

1. Lord Luthor Tyrell's widow Lady Olenna Tyrell nee Redwyne
2. Lord Halys Hornwood's widow Lady Donella Hornwood nee Manderly
3. Lord Eddard's widow Lady Catelyn Stark nee Tully

There are however no fixed rules about names of married women. We see some ladies who take their Lord Husbands' names and then we see some who retain their own names. Queens as a rule, never take their husband's name. Of course in case of widows, it appears they may carry their dead Lord's name as long as they do not marry again. For example, Catelyn would be Catelyn Frey had she married Lord Frey after Ned's death.

"It must happen," said Catelyn, though not gladly. "I have no more wish to suffer Walder Frey's insults and complaints than you do, Brother, but I see little choice here. Without this wedding, Robb's cause is lost. Edmure, we must accept."

"We must accept?" he echoed peevishly. "I don't see you offering to become the ninth Lady Frey, Cat."

"The eighth Lady Frey is still alive and well, so far as I know," she replied. Thankfully. Otherwise it might well have come to that, knowing Lord Walder.
_{ASOS - Catelyn IV}

And Donella Hornwood would become Donella Manderly if she married one of the Manderlys:

Midday came and went. Maester Luwin sent Poxy Tym down to the kitchens, and they dined in the solar on cheese, capons, and brown oatbread. While tearing apart a bird with fat fingers, Lord Wyman made polite inquiry after Lady Hornwood, who was a cousin of his. "She was born a Manderly, you know. Perhaps, when her grief has run its course, she would like to be a Manderly again, eh?" He took a bite from a wing, and smiled broadly. "As it happens, I am a widower these past eight years. Past time I took another wife, don't you agree, my lords? A man does get lonely." Tossing the bones aside, he reached for a leg. "Or if the lady fancies a younger lad, well, my son Wendel is unwed as well. He is off south guarding Lady Catelyn, but no doubt he will wish to take a bride on his return. A valiant boy, and jolly. Just the man to teach her to laugh again, eh?" He wiped a bit of grease off his chin with the sleeve of his tunic.
_{ACOK - Bran II}

Matrilineal Marriages

There is no such thing as a matrilineal marriage in Westeros.

Children born of a marriage can take name of either one of their parents, overwhelming majority prefers their father's name. However if it's the mother who is a landed noble and father who's a mere consort, Children take name of their mother e.g. Princes of Dornes.

A man can change his surname to another dynasty's if:

1. He is married to a woman of that dynasty
2. He is descended from a woman of that dynasty

For point #1, we see King Joffrey Lannister, born as Joffrey Lydden. His wife was the only daughter of King Gerold III Lannister. When Gerold died, it was Joffrey who took the throne as Joffrey Lannister by right of marriage. That's when House Lannister changed from First-Man stock to Andal stock.

In time, Lannister kings wed their children to Andals as well; indeed, when Gerold III died without male issues, a council crowned his only daughter's husband, Ser Joffery Lydden, who took the Lannister name and became the first Andal to rule the Rock.
_{TWOIAF - The Westerlands}

For Point #2, we see examples of Beren Tallhart. Beren was son of Berena Hornwood, sister to late Lord Halys Hornwood. When Lord Halys died with heir and only son therefore extinguishing House Hornwood in male line, Beren's father proposed to Winterfell that if Beren is named heir to Hornwood lands, he will take Hornwood name so that the house could continue.

"I will, ser," said Leobald, and only then raised the matter of Lady Hornwood. Poor thing, with no husband to defend her lands nor son to inherit. His own lady wife was a Hornwood, sister to the late Lord Halys, doubtless they recalled. "An empty hall is a sad one. I had a thought to send my younger son to Lady Donella to foster as her own. Beren is near ten, a likely lad, and her own nephew. He would cheer her, I am certain, and perhaps he would even take the name Hornwood . . ."

"If he were named heir?" suggested Maester Luwin.

". . . so the House might continue," finished Leobald.
_{ACOK - Bran II}

Other than facts, there are also Wilding traditions suggesting that House Stark's senior male line died long ago and current Starks are descended from a female Stark and a wildling King Bael the Bard. As per their story, Lord Brandon Stark fathered no children except a daughter. The daughter was supposedly abducted by King Beyond the Wall, Bael the Bard. Brandon was distraught because this meant extinction of House Stark. But later the girl was found with a child fathered by Bael. That child was brought up as a Stark and succeeded his grandfather upon his death. For details, click here.

Answer 2

Yes; Lady Dustin kept her name after her husband was killed, as did Catelyn Stark and Lysa Arryn.

It all depends on the Lady and the Lord who's name she had originally taken. For example, Cersei never took Robert Baratheon's name to begin with, so after his death it was a bit of a moot point.

More info about that here: Why is Cersei commonly referred to as "Cersei Lannister" instead of "Cersei Baratheon"?

In general though, yes, they do keep their husband's name.