I would give illusion/dark a go as they sort of fit thematically and you will have a plethora of pets to play with.
Phantoms army
terrify ghost thing (sorry can't remember name)
phantasm
fluffy
Patron pet
lore pets
This combo is arguably more effective than dark/dark
Time is kind of like a middle ground between /rad and /dark. It's pretty focused on defense/control, but it actually does a lot of the same things as rad. On my dark/time it felt kind of like overkill, while on my ill/time it rounds out the build by making up for illusion's lack of hard control (it's turning into one of my favorite characters, and I haven't even put filled distortion field with procs yet).
For having fun with your kids I'd probably recommend /kin because the extra endurance+recharge would let them have more fun with their characters (they can use their favorite powers more often), and everyone loves fulcrum shift.
Thanks for the ideas/feedback. 
@ Two5boy: I hear you about Ill/Dark, but I don't know that the fundamentally chaotic nature of Ill would mesh really well with supporting the kids? My daughter is the tank, and my son is currently playing a Bot/Storm MM. The Bots would be fine with the chaos, but it would make Freezing rain kinda useless.
@ Vimes_NA: ::nods:: I think you may be right. It's really about enabling THEM to have fun, and Kinetics does that really well. They'd probably get a kick out of things that I usually wouldn't bother with like Repel: Mob golf anyone?! 
Great insight about /Time, I appreciate it. It's a set that I'm interested in, but just looking at it it seems pretty control heavy which Dark/ really doesn't need.
I have to admit, as a LONG time Kinetics player (my original main Argentae was a Mind/Kin that I spent a fair amount of time soloing back in I-2 when it was really sloooooow to solo a controller ), I'd enjoy being a /Kin again.
Kinetics lacks a -ToHit or +Def to pair up with the -ToHit from Dark/ though so the kids characters will have vital roles to play, Kinetics will just make them better at what they do. Lord knows I'll be able to keep the tank upright through almost anything! And as She's WP/, the -ToHit from RttC should pair nicely with the opening -ToHit from Dark/.
Hmmm...
Arg
If you are looking to design a character intended to support your kids' characters, then it really matters what your kids are playing. You mentioned that your son is playing a Bots/Storm. You said your daughter is playing a tank, but you didn't mention the powersets. (I have some experience in making a team to play with kids, so I understand what you are doing.)
The reason I think it would be useful is that you can choose your character to supplement their strengths. Some Tanks are Resistance-based (Fire, Dark), others are Defense based (Ice, Shield), and others are a combo (WP, Invul). If you have defense, is it positional (like Shield or SR) or typed (like Ice or Invul)? Does the tank need some help with her defenses/resistances? Certainly the Bots/Storm could use more Defense to go with the bubbles that the Bots give out, but Resistance could help, too. How much will the kids, with the way they play, need to be healed?
What is your goal when teaming with the kids? Do you lead, or let them lead and follow them to support and buff them? Do you need heals? Do you need Defense or Resistance shields? Do they run out of endurance? Do you need a Rez power?
How is their damage? Do you need to add damage from your own attacks (maybe a Fire/Dark?), or do you need to buff damage (Dark/Kin, which should be a pretty nice combo), or can you let the kids do most of the damage? That might determine if you need Damage buffs or Resistance Debuffs, or if you need attacks yourself.
Robots have Forcefield defenses, but Storm does not have a defense buff. If your Tank could also use a Defense boost, then you may want to look at secondaries that can buff Defense -- but keep in mind whether the defense is positional or typed. Cold might be a good secondary since it has great debuffs and a fantastic Recovery/Endurance buff, too. Time and Dark have some nice powers in these areas, but keep in mind how late in the set these powers might be (as kids often lose interest in things after a while). There is always FF if the kids need defensive shields in the low levels, but it is not a great secondary otherwise.
If you need a combination of heals and Resistance buffs, you could consider Thermal as a secondary. If the Tank has plenty of Defense, then Resistance might be a better choice. Or if the Tank is Resistance based, then more Resistance is usually a great idea. So Dark/Therm might be a choice.
How good are the kids at developing and staying with a strategy? If they play chaotically, then Storm is a pretty good secondary, but more than one stormy adds a LOT of chaos. If your kids are pretty good at playing with strategy, then a second Stormy may not be a good idea. (I have rolled up a Dark/Storm simply because I love the concept. Have you ever heard of the classic Bulwer-Litton sentence used as an example of bad writing, "It was a dark and stormy night."? I made "Dark Stormy Knight." I haven't played him yet, though.)
I would suggest that before you choose your character's powersets, think about what the kids need to be better. I presume that the main goal of this character is to help the kids have more fun playing, rather than to be an optimal character for yourself. There are a lot of combos I would play as a pure support character over a standard character that I planned to level up with the usual mix of teaming and solo.
LOCAL MAN! The most famous hero of all. There are more newspaper stories about me than anyone else. "Local Man wins Medal of Honor." "Local Man opens Animal Shelter." "Local Man Charged with..." (Um, forget about that one.)
Guide Links: Earth/Rad Guide, Illusion/Rad Guide, Electric Control
So, Shadow Fallen (Dark/Dark) is well into the incarnate content, and is getting so effective that it's no longer a good fit for playing with my 9 year old daughter and 10 year old son.
I'm pondering a new alt for playing with them while I keep working on the Dark/Dark incarnate, because it's THAT great a combo.
I'm really tempted to go with a Dark/? to try out how amazingly effective Fearsome Stare is with other secondaries.
The three main ones I'm considering are: /Kin, /Storm or /Time. Rad is also a possibility but it doesn't seem to have quite the synergy that the others have?
/Kin: Fearsome Stare is one of the best FS setup powers in the game. Dark/ would help tremendously with the fundamental fragility of /Kin, while /Kin would boost the damage output of Dark/ enormously, while providing first rate team buffs.
/Storm: The -ToHit and positioning capabilities from Hurricane would mesh REALLY well with Fearsome Stare, and the -Def in Freezing Rain would be an much welcomed debuff. In the late game, the ability to have multiple Tornadoes out + Lightning storm would really amp up the single hard target damage that the set could provide.
/Time has a lot of synergy with -ToHit as well bringing a lot of +rech to the table in the late game. This is the only Secondary I haven't played with enough to have a good feel for the synergies with Dark/
/Rad is always a possibility as it's such a well rounded set, esp with the -Regen it brings to the table.
So for those that have tried out some of the other Dark/ combos, what has you experience been? What would you try if it was your choice, and why?
Thanks in advance for your thoughts and insights...
Arg