Blue stars are notorious for burning too bright and living too briefly for life to develop around them. But is there some way that life could survive long enough to reach the sentience stage? I'm thinking about a terrestrial world like Earth, not an ice moon.
Binary Star Merger - It wasn't always a blue star
If it started with two lower-mass stars, it would allow a much longer history. It would pass through a "peanut-shaped" contact binary phase where the stars share an atmosphere. Most such binaries are stable, but binary mergers are common enough that we have witnessed a few -- though they were more explosive events than you would hope for. Still, we have evidence that mergers could happen slow enough to produce a stable blue star, called "blue stragglers." This is where a cluster of older smaller stars has some blue ones that shouldn't be there, likely formed after the fact from older fellows. If it happened really slow and steady, I suppose you might get a blue star without too much trauma to the planetary system. A rare occurrence, but in the realm of possibility.
Here are a couple sources:
Blue stars live around 10 million years.
According to the most recent timeline of human evolution, homo has developed in the last 2.5 million years, while 10 million years is what it has taken for earlier apes to differentiate until where we are today.
Though making a statistic out of a single data point is never a good practice, I hope you agree that the earlier apes didn't jump out of the pre-biotic soup, but required some billion years of evolution to reach that point.
The only way to maybe shorten the needed time is to have complex life seeded from outside on a terraformed planet, so that some corners can be cut and life evolution gets a head start.
Sentient Life Could Colonise The System
Ten million years is an eyeblink in evolutionary time. However a technological civilization could settle the place. It might even be possible to stabilise the star, though the most likely route we'll ignore would be to use star lifting to turn it into many smaller stars. One (reading ai-gods) could mess with the structure or fusion activity inside the star to make it stable for longer. If that happens, local life might evolve.
Rogue planet capture
Life evolving on a blue star is just not likely. Life arriving there skipping the hard part would be a very convenient event however how can that happen?
Rogue planets are planets with absolutely no star that they orbit. Because of that they are very cold. However, geothermal heat could cause life to exist on them. This is similar to how some theories say life evolved on earth, deep sea geothermal vents. Thus a planet could have life on it, but be rogue and then be captured by a blue star, warming it up. This would cause life to skip ahead to a more reasonable challenge of becoming intelligent with all the groundwork already set.
The real question is how far ahead could life get. Obviously it couldn’t get to the surface, so only aquatic life until reaching the star. The orbit of this planet would also likely not be very good for life. These issues can probably be handwaved away pretty reasonably however.
On a super-earth.
To have life evolve very quickly you need ample resources, and a lot of chances.
So, suppose this star caught a rogue superearth in its habitable zone. This super earth has a thick atmosphere with lots of compounds that block radiation, a powerful magnetic field that re-directs radiation, and a lot of water on the surface.
It also has some dormant life from its previous star. It took thousands of years to cross the stars, but some bacteria and small organisms near thermal vents survived the travel.
Once it reached the star, the massive size and the vast amounts of light from the star let life explode. The planet is 10 times the size of the earth, and due to an extensive set of coral reef like thing in the oceans, has over 100 times the surface area of the earth.
This means life evolves 100 times faster. What took earth 500 million years, this life can do in 5 million years.
Consider that to have life you'll likely also want a habitable planet. During the planets formation a lot of energy is introduced via its constituent mass falling into the planets gravity well - you have a ball of molten rock that continues to collect enough debris to keep the surface uninhabitable by any standard for likely longer then your star is alive.
Even if your planet cools down it still needs a suitable atmosphere. The "Great Oxygenation Event" on earth took an order of magnitude longer than the star. That's no coincidence as the whole surface of the planet is ready to react with any components that aren't inert. But you need them to power a metabolism fast enough to speed run evolution into life forms that are complex enough to gain advantage from sentience and perhaps even sapience.
It seems the best chance for the development of life would be after the death of the star, akin to the question whether life on a rouge planet or on a large terrestrial moon around a surviving gas giant.
The very event of the spontaneous appearance of life is extremely unlikely, since the process of evolution from the "primary soup" into the simplest forms of life is quite long in itself. Evolution from the simplest to intelligent is faster, but also long, and probably will not have time to reach intelligent life forms in the lifetime of the blue star.
For what you are talking about to happen, most likely someone from the outside had to contribute to the acceleration of development, or brought life in some form to this planet.
How about the life developing around a Sun-like yellow star and a near multi-body encounter with the big blue star at some point, getting the planet in a sufficiently wide near-circular orbit around the new star?
The previous star may get ejected or get a different orbit (inside or outside of the new orbit of the planet).
All of these variants are possible with the usual orbital mechanics.