Second-hand Fluke 114/115/117
That's an option, but a bit above the set price range for delivering to Northern Europe: $105+ shipping $48 + import VAT $25 = $178

Although Fluke 17B+
http://www.ebay.co.uk/itm/Fluke-17B-Auto-Range-Digital-Multimeter-/331794146409 might fit the bill. I suppose its backlight function doesn't differ from Fluke 114/115/117?
Grab an older Fluke bench meter with LED display. You get digital you can read from a distance, and in dim light. You never have to change batteries either 
Sorry, forgot to mention - I would need it to be portable.
There are plenty of meters available with the ability to customise the backlight options...
Thanks for the extended post, but to be even more helpful could you please name some particular ones which has this ability for given budget?
One example for application for this kind of use would be:
As here in Northern Europe we don't have that much daylight available especially during winter time having a backlight to take readings is rather essential. E.g. when working on a car by the engine and setting up the DMM then going to turn the ignition on to see immediate effect of voltage change/ current consumption/ resistance etc.
These cost $2:
..
PS: If you're working with electronics/multimeters in the dark, aren't you already going to be wearing one of these?
Thanks Fungus, it would help in some situations, but in others it might not. E.g. the light will bounce off of the meter's display when set at certain angle, or it might be difficult to shine the light through windshield
and through the crack of an opened hood.
The Fluke 87V has no backlight timer.
...
Time to pony up some cash.
Aaah - of course they've done it correctly and the 87V would be perfect, but that price tag though.

However I found out that
Dave's BM235 Multimeter has
exactly what I need - according to manual - see attachment:
Let's just hope that even with the backorder the price for shipping and import VAT will not skyrocket.

I'm still open for more suggestions.