Gee, I wonder how tragic deaths might have brought two grieving families together?

Prince
…Capulet! Montague!
See, what a scourge is laid upon your hate,
That heaven finds means to kill your joys with love.
And I for winking at your discords too
Have lost a brace of kinsmen: all are punish’d.
Capulet
O brother Montague, give me thy hand:
This is my daughter’s jointure, for no more
Can I demand.
Montague
But I can give thee more:
For I will raise her statue in pure gold;
That while Verona by that name is known,
There shall no figure at such rate be set
As that of true and faithful Juliet.
Capulet
As rich shall Romeo’s by his lady’s lie;
Poor sacrifices of our enmity!
Prince
A glooming peace this morning with it brings…

(Incidentally, don’t really agree that they blame him for her suicide. It’s more complicated than that.)

Here’s how I think it went down: 

1. Ned and Ashara were lovers at Harrenhal – after all, they’re both young, neither of them are betrothed to anyone, so there’s no dishonor as Harwin tells us. Indeed, Ned may have even asked for Ashara’s hand or been in the process of negotiations for a betrothal – he’s the second son of a Great House, she comes from a Lesser House but an ancient one, no one’s an heir so that’s less of an issue, it’s a good match – when Lyanna disappears and Brandon rides to King’s Landing.
2. The young couple are divided by civil war – his father and brother have been murdered and his sister’s abducted, he’s running for his life, she’s looking after Elia Martell in dangerous circumstances (although GRRM says she’s not nailed down so perhaps they have time for one more reunion before he gets married), Rhaegar’s missing in scandalous circumstances. 
3. Ashara becomes pregnant and goes home to Starfall for her lying in, except that the civil war and Ned’s forced marriage prevent the child from being born in wedlock. This is much less of an issue in Dorne, remember. She has a stillbirth. 
4. Ashara finds out that her brother is complicit in the abduction and imprisonment of the sister of the man she loves at the orders of the man who humiliated her mistress and provoked the civil war that ruined her life, when Arthur writes home asking for the family to send Wylla to the Tower of Joy because Lyanna is pregnant and needs a midwife for her secret royal baby.
5. When it’s clear that the war is over, Ashara sends word to Ned where he can find his sister. 
6. The fight at the Tower of Joy happens. Soon after, Ned arrives at Starfell with Dawn, his nephew, and his sister’s body. The combined grief of her stillbirth, her culpability in the death of her brother, and everything else leads Ashara to commit suicide.

At this point, think of the shared grief of the two families. Ned’s lost his father, his brother, his sister (in part because of Arthur), the woman he loves, their child, and almost all of his friends. The Daynes have lost Arthur, Ashara, and Ashara’s child. The mutual recognition of the common tragedy that has befallen them leads to reconciliation. 

After all, Ned stays in Starfall for some time, because newborn Jon isn’t about to travel. The family gets to know Ned and move past initial reactions. They see the way that he’s agreed to protect this innocent, helpless child at the expense of his own reputation, and they agree to do the whole coverup with Wylla to shield both Lyanna and the child, but also in a way to preserve the reputation of both Arthur and Ashara. 

And so the Daynes remember Ned as a tragic figure, the man who might have become their good-son/brother if it hadn’t been for the war.