What do we learn from Gertrude's farewell to Ophelia?

The short answer is that we learn either that

(1) Gertrude had been prepared to accept Ophelia as her daughter-in-law; or

(2) regardless of whether or not Gertrude had been accepting of this prospect, she was content to let the mourners at the burial believe it.

We also learn that Gertrude either believed Ophelia to have died a virgin, or wished her listeners to think she believed Ophelia died a virgin. To understand the significance of these points, we need to take the context into account.

Midway through the first scene of Act V, Hamlet is shocked to learn Ophelia is dead. While hanging out in the graveyard, he sees the procession of mourners, and overhears his mother, Gertrude, speak this farewell to the dead Ophelia:

"Sweets to the sweet: farewell!

(Scattering flowers)

I hoped thou shouldst have been my Hamlet's wife;
I thought thy bride-bed to have deck'd, sweet maid,
And not have strew'd thy grave."

Thus, Gertrude indicates her hopes that Ophelia had lived to marry her son, and suggests that Ophelia died "a maid" (meaning a virgin).

In Act II, Scene 2, Polonius informed Gertrude and Claudius that Hamlet had been sending love letters to Ophelia. Polonius—who was Ophelia's father—made it clear that he told Ophelia not to expect marriage.

"Lord Hamlet is a prince, out of thy star;
 This must not be."

Polonius wanted the King and Queen to understand that Polonius knew his place, and had no unseemly ambitions to promote his daughter in this way.

It's not clear (at this earlier state in the play) what Gertrude thinks about a prospective marriage between Ophelia and her son. A bit later, she tells Ophelia that she hopes Hamlet's bizarre behavior is attributable to his alleged love sickness:

"And for your part, Ophelia, I do wish
That your good beauties be the happy cause
Of Hamlet's wildness: so shall I hope your virtues
Will bring him to his wonted way again,
To both your honours."

This hints that Gertrude might have been amenable to Ophelia as a daughter-in-law, but it's by no means certain. Elizabethan audiences understood that princes kept mistresses. Accepting Ophelia as her son's love interest is not the same thing as accepting her as his wife.

Finally, it should be noted that the men in Ophelia's family were worried about the possibility of Ophelia becoming sexually involved with Hamlet. Hamlet's attentions to Ophelia could create gossip about this; Gertrude's comments suggests she rejects the idea that Ophelia had lost her virginity.