I'll tell you why I have a problem with giving it too many options. First, the typing is amazing both defensively AND offensively and second, the stat spread is awesome for a bulky pokémon.
So, if we decide to overload it with a lot of moves, we are risking making it too powerful. And you may say "hey, if it's too powerful after testing it, let's just tweak it a bit". The problem is, there will be A LOT of bitching and moaning about nerfing it, even if it is necessary. It will be easier to give it more options if it seems limited than the contrary.
Anyway, my main reason to not give it any recovery move besides Rest is to, again NOT make it outclass Dusknoir. Dusknoir is about as sturdy as the mummy, but has less offense and a bad recovery move. Plus, a worse typing. The only thing that is going to save poor old Dusknoir from sinking to that pokémon limbo known as BL (where other poor outclassed pokémon are, like Blaziken) is the access to WoW, a non-rest recovery move, and a somewhat deeper movepool. And even them, I'm sure it will fall in usage.
Stuff the mummy with STAB otions: Mach Punch, Karate Chop, Drain Punch, Brick Break, Hammer Arm, Cross Chop, Astonish, Shadow Sneak, Shadow Punch... Give it some decent support movepool: Rest/Talk, Trick Room, Heal Block, Taunt, Bulk Up, Substitute..., and then some other typed move, but not that many: Ice Punch, Rock Slide, Rock Tomb, maybe Power Whip...
The way I devise the mummy is as a bulky attacker, Adamant nature, with 252 Evs in HP, lots of attack EVs and maybe some investment in defense, taking advantage of its good attack, sturdy defenses, and awesome typing. It doesn't need to be a wall, like Dusknoir, let's make it occupy another role, like Swampert.