Beauxis makes France return after six years
The 32-year-old Beauxis was only added to Brunel's squad due to a knee ligament injury to teenage flyhalf Matthieu Jalibert in last weekend's 13-15 defeat to Ireland.
The Lyon halfback's last appearance for France was in March 2012 against Wales.
Brunel said his decision had been influenced by the famously poor Edinburgh weather expected at Murrayfield.
"We think the weather won't be very good and so his experience and kicking game will help us relieve the team if the pressure is a bit strong," said Brunel.
Beauxis is known for having a particularly hefty boot and being a reliable place kicker: he has amassed more than 2 000 points during a career that has seen him win the Top 14 title with giants Stade Francais and Toulouse.
"We know his talent, his game management and individual abilities.
"He hasn't always shown them at the highest level, he hasn't been consistent.
"Since the start of the season with Lyon he's played very well. He's full of confidence, calm and offers assurance."
Jalibert, 19, had widely been expected to be replaced by Anthony Belleau, 21, who played 50 minutes against Ireland after coming off the bench.
However, Belleau missed a crucial late penalty that ultimately cost France as Johnny Sexton won the game with a last-gasp drop goal three minutes beyond the final hooter after an incredible 41 phases of play.
"I told him why [he's still on the bench], he's understood," Brunel said of Belleau.
"I don't think he's disappointed, he understands why we've picked Lionel Beauxis."
Beauxis made his France debut in 2007 against Italy in the Six Nations and was included in their World Cup squad later that year.
Having begun the tournament as replacement, he was the starting flyhalf in France's memorable quarterfinal victory over New Zealand, and then again when the tournament hosts lost to England in the semifinals.
He is one of three changes to the starting line-up, all injury induced, while powerful No.8 Louis Picamoles is one of three new faces on the replacements bench.
Marco Tauliegne comes in for Kevin Gourdon, who has an ankle injury, at No.8 with fit-again Geoffrey Doumayrou replacing Henry Chavancy, who has not taken part in contact training this week due to an eye socket injury, in the centres.
Scrumhalf Baptise Serin is in on the bench in place of Antoine Dupont, whose season is over following a knee ligament injury suffered during a nine-minute cameo against Ireland, while Eddy Ben Arous replaces Dany Priso as substitute loosehead prop.
Picamoles was a shock omission from Brunel's original squad to face Ireland with the coach claiming it was due to poor club form. But Gourdon's injury, which sees Tauleigne promoted from the bench, will give the hard-running Picamoles a chance to make an impact in the second period.
Montpellier's Picamoles had responded to his original exclusion in the best possible way, putting in an impressive performance in a Top 14 victory away to champions Clermont two weeks ago.
"I told Louis his performances at the start of the season did not correspond to the talent we know he has," said Brunel.
"He showed last weekend that he wanted to demonstrate his qualities again."
France: 15 Geoffrey Palis, 14 Teddy Thomas, 13 Remi Lamerat, 12 Geoffrey Doumayrou, 11 Virimi Vakatawa, 10 Lionel Beauxis, 9 Maxime Machenaud, 8 Marco Tauleigne, 7 Yacouba Camara, 6 Wenceslas Lauret, 5 Sebastien Vahaamahina, 4 Arthur Iturria, 3 Rabah Slimani, 2 Guilhem Guirado (captain), 1 Jefferson Poirot.
Replacements: 16 Adrien Pelissie, 17 Eddy Ben Arous, 18 Cedate Gomes Sa, 19 Paul Gabrillagues, 20 Louis Picamoles, 21 Baptiste Serin, 22 Anthony Belleau, 23 Benjamin Fall.
Date: Sunday, February 11
Venue: Murrayfield, Edinburgh
Kick-off: 15.00 (15.00 GMT; 16.00 France Time)
Referee: John Lacey (Ireland)
Assistant referees: Nigel Owens (Wales), Paul Williams (New Zealand)
TMO: Rowan Kitt (England)