Around 1990 it was thought that computer architecture needed to switch from Silicon-Oxide to Gallium Arsenide. Seymour Cray even built the Cray-3 that way. I think the reason it wasn't pursued was manufacturing problems. But why did the need for GaAs go away? What technology solved the problem it was supposed to address?

This is a great question, thanks.

GaAs didn't go away. It just didn't go into mainstream cheapo electronics, it stuck with the niches that silicon just can't function in. And those niches are quite plentiful even in consumer devices.

Ultimately the argument in any large-scale manufacture of semiconductors is an economic one. GaAs is expensive, silicon is cheap. If you can make it good enough out of silicon, don't bother trying to do it out of GaAs.

The two biggest bonuses to Gallium Arsenide over Silicon in terms of device performance are:

1) Much higher Electron mobility -- that is, you can shoot electrons across a transistor faster, get higher gain and higher bandwidth out of transistors, and get better n-type conduction with a lower doping level.

2) Direct band-gap, allowing for direct on-chip generation of light for signaling and detection, which would have effectively allowed for all-optical buses instead of the criss-crossing network of metal lines.
Direct vs indirect band gap, by Silicon Nanocrystals

IBM's new chip using Silicon photonics:integrate optical communication technology
This stuff probably would have come at least a decade ago had GaAs won the semiconductor war.

Apply the right technology, and GaAs can whoop silicon's butt. And in fact there are a number of technologies where GaAs is applied that silicon technology just can't stand up to. For instance, RF/Microwave amplifiers. In consumer electronics, a GaAs-based Heterojunction bipolar transistor is likely at the heart of your high-end WiFi receiver and transmitter, and is almost certainly at the heart of your cell phone's receiver and transmitter.