[nudephil (OP)]

Here's a question Donald sent in via the webform:

During a solar eclipse, the moon is between the Earth and sun. The Earth wins this tug o' war with the sun and the moon continues to orbit the Earth. However the moon is slowly drifting farther from the Earth. At what distance and when will the moon be equally attracted to both the sun and earth and what will happen!

Can anyone speculate?

[evan_au]

The Moon's tides are slowing down the Earth, When the Earth rotates more slowly, the Moon's recession also slows down.

Calculations suggest that if things continue as at present, the Earth's rotation would slow down, and would become tidally locked to the Moon in about 50 billion years.

However, the Sun is scheduled to go Red Giant in about 5 billion years, and will likely melt the Earth and Moon. So the Sun won't pull the Moon away from the Earth (if things continue as at present).
See: https://en.wikipedia.org/wiki/Orbit_of_the_Moon#Tidal_evolution

[Halc]

During a solar eclipse, the moon is between the Earth and sun. The Earth wins this tug o' war with the sun and the moon continues to orbit the Earth.

The Earth never pulls as hard at the moon as does the sun.  Hence the path of the moon is always curved towards the sun, even during an eclipse.  This does not break the orbit about the Earth since the Earth is similarly pulled towards the sun.
What would break the moon away is if the solar tidal force acting on the Earth/moon system is greater than their mutual attraction to each other, and that's nowhere close.

[Janus]

What would break the moon away is if the solar tidal force acting on the Earth/moon system is greater than their mutual attraction to each other, and that's nowhere close.

This distance is determined by what is known as the Hill Sphere, which for the Earth is ~1.5 million km, or 3.9 times the present distance of the Moon from the Earth.  At its present rate of recession, the Moon would take some 28 billion years to get this far away.  As already mentioned, as the Moon recedes it does so at the expense of the Earth losing its rotation.  The Earth presently slows by a rate that has the period of rotation increase by 1.7 milliseconds per century.  At a distance of 1.5 million km, the Moon would take some 211 days to orbit.  For the Earth to become tidally locked to the Moon it would also have to rotate in 211 days. At its present rate of slowing, this would take just over 7 billion years. 
Now evan-au gave a figure of 50 billion years for the Earth to tidally lock to the Moon.
This number almost twice as large as the number given for the time to reach the Hill sphere distance, so at first blush, one might be tempted to think that the Moon would escape the Earth before tidal lock could occur.
However, it is not that simple.  The value for the time it would take to recede to the Hill sphere that I gave above assumed a constant rate of recession, as did the time needed fro the earth to slow its rotation.  In realtiy they won't be constant.  The tidal braking that drives both of these is heavily dependent on the distance between Earth and Moon. All other things being equal, it varies by the distance to the power of 6 ( doubling the distance decreases the braking by 1/64) So both rates will decrease over time.  It could take more than 50 billion years for the Moon to reach the Hill sphere distance.
But, even here, I said "all other things being equal".  They won't be.  There are two other factors that determine tidal braking. the Disipation factor of and the Love number, and while we have a good idea of what these values for the Earth are now, they can change with time.   For the Earth, a number of factors figure in when determining these values.  They involve both the ocean and Earth tides.   Ocean tides provide a portion of the tidal friction.   As the billions of years go by, the Earth will lose them, But even before that, the shape and distribution of the continents play a role,  As they continue to drift, the ocean tidal friction will vary.     The Earth tide depends on how "pliable" the Earth is.  As the Earth ages, we would expect its interior to cool, the crust thicken and for it to flex less under tidal influence.  This also has an effect on tidal braking.   So even the 50 billion year time scale for tidal locking is a bit of a ball park figure as it is calculated assuming "all other things remain equal".

Of course the Sun will have gone through it's cycle of expansion to red giant, ejection of a planetary nebula, and cooling back to a White dwarf long before any of the above could take place.   Even if the Earth were to not be engulfed by the expansion, drag from the red giant's chromosphere will likely cause it to spiral into the Sun over time. 

[Petrochemicals]

The moon and earth are both in orbit around the sun, around a common centre of gravity. The moon pulls the earth backward and forward, side to side from a view point of this centre. The drift is further from the sun one side and closer the other. The point of equal attraction will probably be variable as to whether the moon is in a prograde motion or not and which side of the earth the planet the moon was,  the moon would enter an unstable orbit of earth.

[Halc]

The point of equal attraction will probably be variable ...  the moon would enter an unstable orbit of earth.

There's no probability involved.  The point of equal attraction is an orbital distance of about 2/3 of the current distance between Earth and moon.  Orbits at that low altitude are quite stable.

variable as to whether the moon is in a prograde motion or not and which side of the earth the planet the moon was

None of these factors make any difference to the attraction, period, or immediate stability of an orbit.  The attraction is a function of mass and separation distance, and not at all on its motion.  A moon in prograde motion will be equally attracted (and have the same orbital period) as a similar moon in retrograde motion.  The difference might be in that the tides might (not always) slowly add to the orbital energy of a prograde orbit, and even then, not in the long term.

[evan_au]

Disipation factor of and the Love number....  They involve both the ocean and Earth tides.

The drag of ocean tides is a large part of this tidal slowing of the Earth & Lunar recession.

In about 2.5 billion years, the Sun is expected to be hot enough to boil Earth's oceans.
- With no liquid oceans, there is no drag from ocean tides
- The tidal drag on the (now very dense) atmosphere is small
- The tidal drag on the solid Earth is very small

So the recession of the Moon is expected to slow significantly after 2.5 billion years.
See: See: https://en.wikipedia.org/wiki/Orbit_of_the_Moon#Tidal_evolution