Fun question! There are two things here:
i) What is the most popular alternative to GR?
ii) What is the best alternative to GR given the detection of these echoes?
I'll address both in turn, but they are interrelated.
What is the most popular alternative to GR?
GR is what's known as a tensor theory. This means that it uses a tensor field to assign a tensor to every point in spacetime. For example at every point around a black hole there is a tensor. The value of these tensors then 'tells' a particle the path to take around a black hole.
The main alternative to this are called tensor-scalar theories. In this case, in addition to the tensor field of classical GR, there is also a scalar field. Analogous to a tensor field, this scalar field assigns a scalar to every point in spacetime. This scalar field is equated to $1/G$, where $G$ is the gravitational constant. Since the scalar field varies from place to place, this is equivalent to $G$ being variable. In classical Einsteinian GR, $G$ is constant. The most well-known tensor scalar theory is Brans-Dicke theory. This theory requires a coupling constant $\omega$ in the field equations, which has a constant value, but that value is unknown and can be altered so as to match experimental observations. Consequently Brans-Dicke theories are hard to falsify which is often the mark of bad explanation and so most physicists stick with Einsteinian GR (for now).
What is the best alternative to GR given the detection of these echoes?
This is where it gets really interesting. The echoes mentioned in the article you linked to arise due to a key priciple of quantum mechanics which is known in fancy language as unitarity, but is better understood as 'information is conserved'.
If we smash together two particles (e.g. at CERN) then gather all the possible information on the particles that result (trajectories, charge, baryon number etc.) we should be able to recreate all the information about the original 2 smashed particles. This is because information is conserved in quantum mechanics. However we run into a problem when we try to apply this principle to black holes.
Stephen Hawking's famous result is that black holes radiate. This is Hawking radiation. A pair of virtual particles pop into existence near the event horizon. One with negative energy is taken into the black hole, whilst the other escapes (see also this physics.stackexchagne answer). Another famous theorem about black holes is the No-hair theorem which says that all properties of a black hole - including the radiation it emits - are determined by just 3 parameters, the mass, charge and angular momentum. This is a problem because it means that Hawking radiation is independent of the sort of material entering the black hole. There is no way to find out all the information about what was thrown into the black hole by getting all the possible information contained in the Hawking radiation. This is known as the black hole information paradox.
A proposed solution to this information paradox is known as a black-hole firewall. The ideas is that when one virtual particle is absorbed by the black hole, the entanglement between the particle pair is broken. This releases huge amounts of energy and results in a firewall around the black hole, just outside the event horizon, that would incinerate any matter that attempts to pass through it. Correlations between emitted particles would then carry the information about the particles that fell into the black hole (note this is quite a heuristic explanation - see the original 'AMPS' paper for a more involved discussion).
The echoes reported in the linked article occur as a direct result of this firewall. The model considers the event horizon as a mirror and the firewall as a partially permeable surface. Gravitational waves would oscillate between these two boundaries, with some gravitational radiation being released each time the wave strikes the firewall, which we then detect as echoes.
The problem with this firewall solution is that it violates the equivalence principle, which we can state in this case as 'the event horizon is not a special place'. According to classical GR, an infalling observer would notice nothing different as she passed the event horizon. However, the firewall solution defies this principle - she would notice something different as she is thermalised into dust by the firewall! The reason this is a problem is that the equivalence principle is a key foundation stone in both Einsteinian GR and Brans-Dicke theories.
If the report of these echoes is true (I note they are still far off the required $5 \sigma$ significance), then it would mean that a firewall exists, and consequently the equivalence principle does not hold globally. That is, clearly the equivalence principle is somewhat right - it has been used to form GR which has passed a plethora of tests in the weak field regime - but perhaps it does not apply to every corner of the universe. Perhaps close to the surface of black holes, where we expect quantum effects to play a more significant role, the equivalence principle does not hold.
The solution will require the development of a full quantum theory of gravity, the two main strands of which are string theory and loop quantum gravity. For discussion on either of these two theories, you can consult existing answers on this very site: Give a description of Loop Quantum Gravity your grandmother could understand or Why String Theory?