mattj and I tested the following this evening:

What is analyze's power boost?

It's almost certainly 1.3x.

In this test, we had a hacked min speed Analyze Magnezone use Thunderbolt vs. max HP Blissey. The normal damage range would be 132-156 HP without Analyze.

With Analyze, the damage was:

187, 174 (crit), 199, 183, 181, 190, 193, 195, 183 (crit), 199, 199, 187, 174

[or a range of 174-199]

(Damage from crits is already halved and rounded down.)

At least we know for sure it's between 1.25x and 1.33x, because the highest damage from 1.25x would be 195 (too low), and the lowest damage from 1.33x would be 175 (too high).

With 1.3x, the damage range would be 171-202 HP.