tl;dr, the orbit of Mars doesn't change
So, since this mass increase isn't going to significantly affect the motion of the Sun (if mars suddenly became the mass of the sun it obviously wouldn't stay in the stay orbit because the Sun would now be moving too and that complicates things) we are able to look to Kepler's third law to tell us that the mass of a body in the system isn't really part of its orbital path. But that might not be too intuitive, so we can derive it from just some basic physics to make it clearer why Mars' mass doesn't really affect its orbit around the sun.
Let's assume the orbit of Mars is pretty circular (this is a good approximation), we know that the acceleration felt by an object going in a circular path (aka the centripetal acceleration) is $$a = \frac{v_T^2}{r}$$ where $v_T$ is the tangential velocity (the velocity of the object that is not towards the center) and $r$ is the distance from the object in circular motion to the center of the circle. Since $F = ma$, this times the mass of Mars $m_{mars}$ is the force felt by Mars. We also know the equation for the force of gravity is
$$F_{gravity} = \frac{GMm}{r^2}$$
so setting these equal to each other we get
$$m_{mars} \frac{v_T^2}{r} = \frac{GM_{sun}m_{mars}}{r^2} $$
we can then solve for $v_T$, noting that the mass of mars cancels out on each side
$$v_T = \sqrt{\frac{GM_{sun}}{r}} $$
we then note that, for a circle, the tangential velocity is the distance ($2\pi r$ since we're talking about a circle here) over the period $P$
$$v_T \approx \frac{2\pi r}{P} $$
substituting this in, we find
$$ P^2 \propto r^3 $$
also known as Kepler's third law, showing us that the mass of Mars doesn't affect how long it takes for Mars to make an orbit, nor how far it is from us, and essentially meaning that nothing noticeably changes in the orbit.
Note
This is assuming more or less that the mass of Mars had always been the mass of Earth as opposed to it increasing suddenly; if it increased suddenly, a mechanism would need to exist for the mass to increase suddenly, and the most realistic scenario for this is a collision, which would most certainly change the orbit of Mars. Even if an advanced civilization had the capacity to bring mass somehow to Marss to increase its size, you could essentially think of a delivery of this amount of mass as functionally a series of smaller collisions that would ultimately add up.
Edit: Let’s further consider the idea of an instantaneous mass increase; while a collision would be more or less an instantaneous mass increase, the direction and momentum of the colliding object would matter a great deal, and so let’s suspend some disbelief and say, somehow, the mass increased without any outside influence (in other words, we treat this as a closed system where momentum is conserved).
Since angular momentum must be conserved in this scenario, $$L_{initial} = L_{final}$$ or in other words $$m_{mars}v_{T,i}r_i=m_{earth}v_{T,f}r_f$$
$$m_{mars}\frac{2\pi r_i^2}{P_i}=m_{earth}\frac{2\pi r_f^2}{P_f}$$
Using Newton’s version of Kepler’s 3rd law
$$P^2=\frac{4 \pi^2}{G(m_1+m_2)}a^3$$
And using the simplifying assumption $m_{sun}>>m_{earth}>m_{mars}$
$$\frac{m_{mars}^2r_i}{m_{earth}^2}=r_f$$
Seeing as the mars is about 10% the mass of Earth, this means that the distance of Mars from the Sun would decrease 100 fold, at least if no other bodies were concerned. It would probably go into some chaotic trajectory due to its interaction with other planets, and from there it would be anyones guess what would happen, although crashing into the sun seems likely.
It’s worth noting that, while we chose to conserve angular momentum, energy here is not conserved because we are magically synthesizing new mass out of nowhere. This situation may seem counterintuitive, and it should, because this scenario isn’t physical, but also because orbital mechanics are notoriously not intuitive.