So, most important requirement first:
Cost is an issue, so the center cannot be built entirely underground. Given the above requirement for geological stability, it may not be possible to build any of it underground. However, no more than 50% of the center may be considered underground for the purpose of this analysis
This is of course, a load of cobblers. If cost was an issue, it would be built on the moon. Or multiple redundant orbital datacentres. Or on an abyssal plain. All of these are extremely difficult to get to, and interfacing with a secure datacentre when you've got there will clearly be beyond the abilities of all but a very small group of organisations.
The Martian option is vastly more expensive and vastly less convenient and doesn't demonstrate a usefully greater level of security than those, so clearly this is a vanity project and money is no object. Lets not pretend otherwise.
Now we've got that that out of the way, the baffling restriction on "no more than 50% of the facility may be underground" is so poorly specified that it must be the product of upper management who are both financially and technically incompetent, as even the most engineering-adverse beancounter would give a much tighter set of restrictions to minimise cost. Given that they don't know what they're talking about, this issue may be worked around to a certain extent.
The facility will be on Mars, which has a little over a third of Earth's gravity, and so little water that digging holes and burying equipment is a trivial exercise compared to shipping a datacentre there. You don't even have to do any environmental remediation; its a dead wasteland. Take a look at this PDF: Radiation Protection Strategy Development for Mars Surface Exploration (PDF). Scroll to page 12 or so, and you start seeing nice diagrams like this:
2 metres of dirt is all it takes. Hell, even a metre will do it at low altitudes. In Martian gravity, that's a pretty light loading for the roof of the facility to support. You'll probably need to send out robot bulldozers to prepare the site anyway, and scraping a shallow grave for your folly is well within their capabilities. Once the facility is assembled, just bulldoze the extracted regolith back over the top in a comparatively thin layer.
I'm sure the same upper management drone will complain about how this is still technically "underground", but really all we're doing is using loose regolith as a coating material... it is no more "underground" than a stone house with a slate roof is somehow "underground" because of what it is made of. I'm sure you'll be able to give them the brush-off, especially when you show them the figures for the reduced cost of shipping shielding from Earth.
The center cannot use Mars-orbit satellites to communicate with Earth.
I have actually sprained my eyes they're rolling so hard. You need to find the pointy-haired imbecile who is clearly trying to sabotage this project and fire them (and at the very least, replace them with someone who knows how to properly derail it rather than making these half-hearted bizarre requirements that can be worked around with a little bit of thought).
You'll be happy to know that my backup plans involve communication mechanisms that can be mounted to Phobos and/or Deimos, and assuming that will be rejected too (because you need to let the PHB reject something to make them feel like they're actually contributing) there's also a proposal with a more conventional constellation of communications satellites than can be placed in halo orbits around the Mars-Sun L1 and L2 Lagrange points to provide continuous coverage of the Martian surface (additional details may be found here: Sun-Mars libration points and Mars mission simulations (PDF))
Nothing need orbit Mars itself. Everything will be great, budget approval will be straightfoward. Working with other national space agencies to co-develop and co-fund this part of the project is clearly possible as it has actual use unlike the main subject of this project.
The budget and design time saved by replacing the Big Honking Antennae with Sensibly Sized Antennae can help fund a more sophicated set of construction robots to facilitate the improved shielding proposal above, and provide a means to transport ice to the facility from elsewhere on the Martian surface. Having communication capability across the entire Martian surface allows the facility to be sited anywhere (not merely places with reasonably frequent lines of sight to Earth), and allows easier teleoperation of construction and support robots.
The improvement in comms link availability (24 and a bit hours a day, 687 days a year, no daily downtime) will increase facility reliability and usefulness and will help drive costs down and customer adoption rates up. The risk of downtime due to satellite failure is low (both satellites would have to go down, vs. the risk of the BHA array failing) and the logistical requirements of running the facility are sufficiently high that bringing backup satellites in shielded coccoons for deployment after a serious CME event would seem to be within the realms of practicality.
The center wants to have access to as much water as possible.
Easy. Don't build it on Mars.
Given that you've already taken the hardest, silliest approach (I mean, if you're being this mad, at least take it to an ice moon or something) then perhaps you'd like to specify what the water will be needed for? You're certainly not going to be doing anything utterly insane like open cycle cooling of your nuclear reactor with steam towers, like you're still on earth, are you? No. You are not. Similarly, the coolant loops within your datacentre are closed systems. You won't be venting steam or leaking water. If you are leaking, then you're not fit to run a datacentre in a city on Earth, let alone anywhere harder to send maintenance folk. Lets be honest, someone accidentally left in a proposal for a swimming pool, didn't they? Just quietly scratch that bit out, and your outrageous water requirements will be rendered manageable.
Now you can site the damn thing almost anywhere, and have robotic rovers go bring you some ice to fill your coolant loops. Even the driest parts of the equator have potential water access available; you don't need to park right on an icecap (and indeed probably wouldn't want to as the weather tends to be poorer there).
The possibility of good building sites coinciding with good sources of water might lead you to consider siting the facility in the northern dune sea... that's fine and the low altitude means a reduced shielding requirement, but the facility will be rendered more vulnerable to dust storms compared to high-altitude sites.
The center wants to be as geologically stable as possible.
Well, Mars would seem to be a good choice for that; Marsquakes are rare, and weak. So long as the facility isn't built on the fringes of the icecaps, or in small craters, it will likely be just fine.
The center wants to be as protected from sand storms and other weather as possible.
The Shallow Grave proposal provides excellent protection to the facility from what little threat the martian elements provide. The removal of the BHA from the plan makes protecting surface equipment fairly trivial, too.
A small ground of sweeper robots may be required in the aftermath of a dust storm, and possibly during one. Although the facility does not require solar power, backup systems may still use it and solar panels will need to be cleaned. More importantly, any heat radiators for the facility's power plant(s) and computers will need to remain fully operational. If the dust forms a thick enough layer to insulate the radiators, they will need to be cleaned off.
The possibility exists of siting the facility on Tharsis; it is known that the highest mountains are above the level of global dust storms. The shielding requirements are higher, but 2m of regolith will be sufficient.
Olympus Mons photographed by Mariner 9, standing above a global dust storm.
The center wants to be as protected from celestial interruption (solar storms, radiation, etc.) as possible.
The shielding proposal above was designed to protect humans (known to be quite sensitive to radiation) from serious and sustained radiation from the sun and from extrasolar space. It will be quite sufficient for this project's needs.
Project management appear to be entirely disinterested in the potentially more serious issue of how to protect the facility from meteorite strike. This underlines my suspicion that this is a vanity project and isn't intended to be practical or sustainable for the long term. A deeper underground data storage facility may be required, but happily this need not be of excessive size and so would fit within the spirit and the wording of the "no more than 50% underground" requirement. Reconstructing the upper portion of the facility (including the communication arrays and cooling systems) after a meteorite strike is apparently outside of the scope of this brief, and will be left to some other poor schmuck. Please note that destruction of the reactor cooling apparatus might not be fatal to the reactor if there was emergency cooling to let the system shutdown cleanly, but a backup plan may need to include shipping out a new reactor if the old one is seriously damaged. I don't like the look of those backup cost figures, let me tell you.
The center must be energy self-sufficient, which means nuclear power
I'm... actually astonished. Presumably the previous PHB was on a coffee break or something and was unable to demand wind or solar power. Please expedite the work on this part of the project before they notice and cancel it; sink sufficient money into it and its inertia will hopefully carry us through this risky period.
A particularly large datacentre will be using less than a gigawatt of power. A nuclear reactor with an output of about 1GWe (electrical, implying perhaps 3GW thermal output) will require about 25 tonnes of enriched uranium a year. The fuel assemblies for these will of course increase the weight somewhat... using the Finnish Olkiluoto reactors as an example, the assemblies add perhaps an additional 50% overhead.
The total freight budget for the project is outside of the current set of requirements, but 50 tonnes a year sounds like it might be a reasonable ballpark figure (to include new fuel assemblies, consumables, lubricants, spare parts that cannot be fabricated on site, secure data shipment and so on). The Earth-Mars synodic period is a little over 2 years, so each actual shipment will need to take over 100 tonnes unless high-energy faster transit options are available. This will add a non-trivial continuous cost.
Smaller reactors will of course require less fuel, but will in turn limit the size of the datacentre. The scale of the operation was unspecified, so the tradeoffs here will have to be left to upper management, god help us.
The practicalities of establishing a fuel reprocessing facility on Mars, or launching highly enriched fuel rods from earth, or mining Martian uranium (we know there is some but disposition of decent ore bodies remains unknown) to produce fuel assemblies in-situ have been considered, and were rejected as being too ridiculous, even for this project.