This is a very broad question, but the general answer is that your mother is unfortunately mistaken. As you mention, the US was involved from December 1941 in a logistically complex and extremely bloody war with Japan in the Far East. But their contribution to the war goes further than that, across several different fronts. Here's a non-comprehensive list of examples:

1. Economic warfare: under the Lend-Lease programme, signed into law in March 1941, the US government provided c. $50 billion in financial and material aid to its wartime allies. Yes, Lend-Lease was partially designed as a kind of 'passive war fighting' — a stratagem designed to balance the need to support US allies at war, while respecting limited public appetite for active intervention (at first) — but it's nonetheless significant as a contribution to the war effort by its sheer scale.
2. The air war: From June 1942 to May 1945, the United States Army Air Forces (USAAF) were engaged in a massive strategic bombing campaign against Germany. Here's a brilliant dataset from the USAAF Office of Statistical Control showing just how many sorties US aircraft were flying on a month-by-month basis. Look at the grand totals: 669,235 sorties flown against Japan from December 1941 to August 1945 — compared to 1,693,565 against Germany in the same period. US bombers dropped 1,554,463 tonnes of ordnance on Germany, compared to 502,781 on Japan. Now granted, that discrepancy is largely due to the tactical complexities of aerial warfare at the time: the USAAF had access to airbases in Britain making air strikes against Germany perfectly feasible — meanwhile, in the Pacific, a major objective of the island-hopping campaign was securing airbases within range of Japan (with the Battle of Okinawa as the ultimate crucible for this). But it's nonetheless an important illustration of 'what America was doing' all that time.
3. The North African and Italian campaigns: from the start of Operation Torch in November 1942, American boots were on the ground in North Africa. Yes, the Americans came relatively late to the North African campaign, but the defeat of German-Italian forces in North Africa led directly into Operation Husky (the invasion of Sicily) and the subsequent Italian campaign — in which US casualties were among the worst of any of the Allied nations involved.
4. The intelligence war: studying the archives of the OSS or reading any of the published history about the activities of the agency in Europe and the Middle East should give some insight into quite how actively involved the nascent US intelligence services were in waging the secret war. Admittedly, there's a school of thought which casts the OSS as 'glorious amateurs' — and it's certainly true that the US was far less experienced than its allies at the practice of espionage — but the OSS' contribution to the intelligence war and various resistance movements was far from small.

So, short answer: American involvement in the war was complicated and controversial (there's plenty to read about the domestic political wrangling and lobbying around intervention in Europe that went on in DC) and yes, Operation Overlord is — in terms of scale — the pinnacle of US involvement in the European theatre, but it's simply not the case that America was sitting idly by watching Europe burn.

As a personal aside: nor do I think there's seriously much sense of that in either popular memory or historical scholarship in Britain (source: I'm a British history graduate.) You still get that old 'Americans turning up late for wars' joke, though...