[Holmes in Holland]

In alkanes, branching leads to decreased molecular overlap and therefore decreased dispersive forces between the molecules, and lower melting points.

In butanols, the only one which is solid at room temperatures is 2-methylpropan-1-ol.  I have searched in my undergraduate textbooks, but not found suggestions why the more branched (and tertiary with presumably less available hydroxyl group for hydrogen bonding?) alcohol should have the highest melting point.

(a high school chemistry teacher trying to make sure he is teaching the correct things!)

[AlphaScent]

You are on the right track with your initial thought on branched alkanes.  This also applies to branched alcohols.  tert-butanol is also a solid at room temperature.

What are some other forces that a hydroxy group can posses to help with its melting/boiling points?

[Babcock_Hall]

With respect to alkanes, melting points and boiling points behave very differently, at least if pentane is representative.  The boiling points of n-pentane and neopentane (2,2-dimethypropane) are 36 and 9 °C, respectively.  This can be explained on the basis that neopentane is more spherical (has less surface area).  However, the melting points of n-pentane and neopentane are -130 and -20 °C. respectively.

[Enthalpy]

Melting points are extremely complicated. I have read no single explanation - not to mention a method with numbers - that gives reasonable figures. This is known. A few teams try to progress on that.

A few elements that seem to work often:
- Polar groups make solids. Hydorgen bonds as well. Sure.
- Symmetric molecules are more solid.
- Branched molecules are more liquid, but that's complicated.
- Numerous isomers make liquids, similarly to eutectics or mixtures
- The relationship with the boiling point is very weak.

As it looks, the ease to solidify depends on each and every detail of the molecules that allows them to stack properly and have many attractive contact points. As stacking can be done with arbitrary position shift, orientation change... between neighbour molecules, the chances to determine it by hand are about zero.

Just one example:
(CH₃)₃CCH(CH₃)₂ melts at -100°C
(CH₃)₃CC(CH₃)₃ melts at +100°C

For your short alcohols, the most important  factor is how the hydrogen bonds can operate among a network of molecules.