What could the fuel pump failure have been that my flight experienced?
The conversation may have been about fuel pumps or fuel transfer valves but the flight attendant was probably told that as a simple explanation - it would not have been the real cause.

Aircraft have multiple systems to pump fuel, transfer, and pipe it so no single failure will affect normal operation. 

For example, pumps have automatic bypass valves so that other pumps will continue the normal flow past the failed pump. No single transfer valve obstructs fuel flow. There are multiple paths to get the fuel to the engines; either directly or by transferring through a completely different set of pipes to another tank then on to the engines.

What likely happened was there was one or more FILTERS that became obstructed. Both ice particles and algae often obstruct filters. But once again all the filters have automatic emergency bypass valves and pipes. However, it is not unheard of for polluted jet fuel containing large amounts of algae to overwhelm the filters. 

[ In my 10 years with Continental Airlines, we had one aircraft engine obstructed by ice particles, but numerous occurrences of algae contamination. ]


  If the filters and/or bypasses either obstructed engine filters, made
  one or more tanks unusable, or reduced the fuel flow, the Captain
  would have chosen an altitude and speed that gave best fuel economy;
  and would have arrived late.


It is worth noting that algae contamination is quite severe in some parts of the world such as Asia and Pacific Island nations. It is common practice for American airliners to carry more fuel than necessary when headed abroad to help "dilute" the bad fuel in other parts of the world.



Simplified Airbus diagram of a typical fuel system showing multiple pumps, transfer valves, and paths to each engine.



B737 diagram (one of the simplest airliners) showing multiple pumps, transfer valves, and paths to each engine.