What will succeed the Arecibo Observatory? Just a few minutes ago, I got a notification from Space.com stating that the Arecibo Observatory will be, sadly, decommissioned due to extensive damage to its structure. So, with the loss of one of the world's largest telescopes, we will be put at a setback in space exploration and astronomy. So, what will succeed Arecibo and when will this happen?
There's no simple answer. In the immediate future, different radio telescopes around the world will pick up the slack in various ways; how that happens will depend on the needs of individual observers and collaborations. Unless someone was to build an identical observatory at the same latitude as Arecibo, with the same frequency range, receiver options and field of view . . . we'll have to spread out the observing. Given that we didn't really expect to lose Arecibo for more than about a year$^{\dagger}$, there are a lot of things up in the air. Insert &quot;probably&quot; and &quot;maybe&quot; into this answer wherever you please.
I do pulsar timing, so I'll talk about what a future without Arecibo might look like from our perspective. My collaboration's two main instruments were Arecibo and the 100-meter Green Bank Telescope (GBT), with roughly half of the observing time at Arecibo and half at the GBT. There aren't firm plans in place for the exact strategy we'll take going forward, but we'll have to decrease our observing cadence at the GBT (i.e. how often we observe certain pulsars) so we can use some of the time to observe pulsars we'd normally observe using Arecibo. So the GBT can take some of the load, but as we only have a finite amount of observing time, that impacts our other observations. We'll likely have to decrease the cadence of observations and the number of pulsars we monitor regularly.
Pulsar timing arrays will also have options with other existing instruments in the immediate future. A key one is the Canadian Hydrogen Intensity Mapping Experiment (CHIME) in Canada. The folks at CHIME will actually be able to observe all of our northern hemisphere pulsars (CHIME Pulsar Collaboration, 2017), which is excellent. In the short-term, CHIME will slightly lessen the burden of losing Arecibo. We've also done a little bit of timing with the Very Large Array, although I don't think that'll be a major player in the future.
That brings us to telescopes currently in the planning stages or under construction. The DSA-2000 (Hallinan et al. 2019) will hopefully see first light in the late 2020s and would likely become a major part of our ongoing observations. The Square Kilometer Array will also be extremely important for radio astronomy as a whole - I don't know whether we in particular will be using it a lot. The SKA should also reach completion in the next decade.
These are just the current potential options from my corner of the astronomical world - and they're quite tentative. Arecibo was our most-used screwdriver in the toolbox, so we've certainly been hit. We'll hopefully be able to somewhat supplement the GBT with CHIME and ideally DSA-2000 a decade from now - so in that sense, those will effectively succeed Arecibo for our observations. In reality, all that means is that we won't lose quite as much of our observing capacity - but it doesn't come close to making up for the loss. Other collaborations and astronomers will presumably increase their usage at the observatories that are best equipped for their observations, depending on frontend/backend requirements and field of view.
It's not going to be fun. We'll be set back. But science will go on. We'll figure it out.

It's worth updating this answer now that the white paper for the proposed Next Generation Arecibo Telescope (NGAT) has been released. The Arecibo Observatory has proposed a phased array of small dishes mounted on a circular platform ~300 meters in diameter (or potentially multiple smaller platforms) which would be angled in various directions to provide reasonable sky coverage. One variant involved 1,112 9-meter dishes, as smaller dishes provide a higher packing efficiency and therefore maximize collecting area. Regardless of the precise configuration, the sinkhole that housed the former 305-meter dish would ideally be the new array's home.
The aim is for frequency coverage from 200 MHz to 30 GHz via four broadband receivers and a frequency-dependent field of view ranging from 6 degrees at 300 MHz to 3.5 arcminutes at 30 GHz, a significant field of view increase over its predecessor by a factor of 500. The dishes would also be equipped with transmitters for radar observations.
Whether the NGAT will be built is a question for which nobody has any answers just yet. It would cost roughly $450 million, which could take money away from other projects unless it received special Congressional funding or support from private sources. Additionally, the setup has not been tested on such a large scale, so there are some technical and engineering concerns (the group did consider but reject single-dish and classical array designs). All that said, the design was put together in an impressively short amount of time, and even if the NGAT isn't build, the scientific considerations that went into it may inspire future designs for replacement telescopes at Arecibo.

$^{\dagger}$ This isn't totally true - a scenario where we lose Arecibo had already been studied in detail. But . . . I think a lot of folks felt that it was more likely that it would be out of commission for a bit but could be salvaged in some way, and hopefully brought back online before too long. This was not the Thursday we were expecting.