During this Reagan administration they refit and re-commissioned the Iowa Class Battleship. It was my understanding that at the time, the concept of the big gun battleship was considered obsolete. If this was the case why not spend the money on some other sort of ship, such as building more missile cruisers, or refitting more World War 2 carriers? Surely navy to navy combat couldn't be the reason, as the Soviet Navy was based mostly on its submarine force.
In one four-letter acronym: NGFS (naval gunfire support).
Big guns were not obsolete; big guns were obsolete for ship-to-ship combat. The shift from the old WWII-style navy to a modern aircraft carrier and small-gun + missile navy had left the USN with 5" guns as the biggest available for NGFS. This is often sufficient, but also often not sufficient against hardened targets. Also, with 1 gun on a ship, the volume of fire is not high.
While NGFS was a key motive for their re-activation in the 1980s, they were significantly modernised, gaining an effective missile capacity, carrying 16 AGM-84 Harpoon anti-ship missiles and 32 BGM-109 Tomahawk missiles (IIRC, the Iowa class was the first USN surface combatant carrying the Tomahawk; the improved Ticonderoga-class ships carrying the Tomahawk-capable Mark-41 Vertical Launch System (VLS), USS Bunker Hill (CG-52) and later, only came into service in 1986.
The main NSFS activity by the Iowa class was the 1991 Gulf War against Iraq. USS Missouri and USS Wisconsin together fired 51 Tomahawks and about 1000 16" rounds. Submarines and Ticonderoga-class cruisers fired further Tomahawks, for a total of 290 fired in that war (about 2000 have been fired at other targets since, the majority at targets in Iraq). Guns (and air support) are not the only option for ships to support troops on land. Missiles can also be used, so NGFS has been replaced in US usage by NSFS, naval surface fire support. Below, I will restrict the acronym NGFS to specifically refer to guns.
NGFS isn't a super-weapon. The limited range of guns means that it cannot be used if targets are too far inland (in principle, extended-range ammunition (e.g., rocket-assisted) can be used, but such rounds are expensive and carry less explosives), and ships can be vulnerable to missiles and mines. Air support (e.g., by carrier aviation) allows the ships to stay further from sure for safety, and can provide support further inland. However, air support can take longer, and is more dependent on weather than NGFS. NGFS is still useful enough, and the limitations of the USN's 5" guns big enough, so that the USN has sought a better solution. In particular, the 6" Advanced Gun System (AGS) was used on the Zumwalt-class destroyers, intended to fill the NGFS niche left vacant by the last retirements of the Iowa-class battleships. Alas for the USN, the Zumwalt class turned into a giant money-eating black pit, and the planned 32 ships turned into 3 (2 in commission, and the 3rd in progress), and the intended NGFS round for the AGS, the extended-range Long Range Land Attack Projectile (LRLAP), was cancelled, so they cannot be used for NGFS. (And even if it wasn't, NGFS at US$1 million per round is an expensive undertaking.)
With the US Navy's NSFS capability still seen as inadequate, there are new proposals for the re-activation of the Iowa-class battleships. The current estimated cost for re-activation is about US$1 billion per ship, similar to the cost for reactivation in the 1980s (about $400 million), adjusting for inflation. Adding to the cost of a new re-activation is the need to make new 16" shells (or at least refurbish old shells, with new propellant). Reducing the cost is that the missile systems installed on their previous activation are still usable. Despite such repeated calls for their re-activation, it does not appear likely to happen in the near future.