'Round here at the nine-dash-five office there is a gal who uses a broken mug for her daily cuppa. The vessel part works just fine but the gripability is severely lacking. Alright, the handle is downright missing. There's just a little hook at the top and a nub at the bottom. Whenever her cuppa is
especially hot, the ability to hold the cup is reduced. Her thinking is that since the vessel is still sea-worthy, there's no point in wasting the earth's precious porcelain to throw this one away. As she is an amateur boxer, we all agree whole-heartedly with her.
My proposal is for an aftermarket attachment to your mug of choice that replaces a broken handle or adds a second one to your beverage container. This device would have slight flexing to allow you clip it over the top lip and bottom edge of your mug. Either a dense sturdy plastic or lightweight metal would suffice. If your morning burst of wake-up was brewed/steeped at an excessive temperature, you could still hold the mug.
For cups that have no bottom lip, a ring would be necessary to place the vessel into before clipping the top part over the rim. However, since there are a myriad of configurations of height, slope, and diameter, this may be too onerous to tackle.
Of course heat need not be your only enemy. On a hot summer day you'll want to pre-chill your pint glass before filling it with a frothy beverage. If the degrees are too low to make holding it comfortable, you have two options: insert a straw and suck or attach a Handle-On. Whilst the former option may have its advantages, it may be a bit daft in a social setting.
At least three sizes would be necessary: Small for a typical tea cup (and possibly a slightly different design to take into account the underslope of the tea cup; Standard for most coffee/tea mugs; Massive for good old pints or quarts (Though I don't know of many pubs that give you a good quart at once anymore). Of course it can be ported in your purse or pocket ("Is that a Handle-On in your pocket or are you just happy to see me carrying these suds to your table?').