Lithium > Alkaline > heavy duty
Get alkaline, best value for money. Lithium will last longer, but probably not as much as 4-6 alkaline batteries that will cost the same (if you make the total).
Also note that the low battery voltage on ut61e shows up at about 5.5v but the meter itself can operate just fine even under 3v, so the batteries you take out of your meter may still work for a few hours to run a clock or something using very low power, with the 5-5.5v the battery has left.
Lithium 9v batteries start at about 8.4v and they're completely dead at about 5.4v so you would get almost all of their energy before the meter says low voltage.
For example, you can see in this datasheet
http://data.energizer.com/PDFs/la522.pdf that with 5mA draw (a bit more than what UT61E draws) the battery will last about 800 hours.
To compare, you have here an alkaline battery datasheet:
http://data.energizer.com/PDFs/522.pdf on page 2 you can see that with about 10mA draw, the battery will go down to about 6v in around 70 hours. So you can extrapolate about 150 hours at 5mA and down to 5.5v...